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import go_benchmark_functions from scipy

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go_benchmark_functions/__init__.py Normal file
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# -*- coding: utf-8 -*-
"""
==============================================================================
`go_benchmark_functions` -- Problems for testing global optimization routines
==============================================================================
This module provides a comprehensive set of problems for benchmarking global
optimization routines, such as scipy.optimize.basinhopping, or
scipy.optimize.differential_evolution. The purpose is to see whether a given
optimization routine can find the global minimum, and how many function
evaluations it requires to do so.
The range of problems is extensive, with a range of difficulty. The problems are
multivariate, with N=2 to N=17 provided.
References
----------
.. [1] Momin Jamil and Xin-She Yang, A literature survey of benchmark
functions for global optimization problems, Int. Journal of Mathematical
Modelling and Numerical Optimisation, Vol. 4, No. 2, pp. 150--194 (2013).
https://arxiv.org/abs/1308.4008v1
(and references contained within)
.. [2] http://infinity77.net/global_optimization/
.. [3] S. K. Mishra, Global Optimization By Differential Evolution and
Particle Swarm Methods: Evaluation On Some Benchmark Functions, Munich
Research Papers in Economics
.. [4] E. P. Adorio, U. P. Dilman, MVF - Multivariate Test Function Library
in C for Unconstrained Global Optimization Methods, [Available Online]:
https://www.geocities.ws/eadorio/mvf.pdf
.. [5] S. K. Mishra, Some New Test Functions For Global Optimization And
Performance of Repulsive Particle Swarm Method, [Available Online]:
https://mpra.ub.uni-muenchen.de/2718/
.. [6] NIST StRD Nonlinear Regression Problems, retrieved on 1 Oct, 2014
https://www.itl.nist.gov/div898/strd/nls/nls_main.shtml
"""
"""
Copyright 2013 Andrea Gavana
Author: <andrea.gavana@gmail.com>
Modifications 2014 Andrew Nelson
<andyfaff@gmail.com>
"""
from .go_funcs_A import *
from .go_funcs_B import *
from .go_funcs_C import *
from .go_funcs_D import *
from .go_funcs_E import *
from .go_funcs_F import *
from .go_funcs_G import *
from .go_funcs_H import *
from .go_funcs_I import *
from .go_funcs_J import *
from .go_funcs_K import *
from .go_funcs_L import *
from .go_funcs_M import *
from .go_funcs_N import *
from .go_funcs_O import *
from .go_funcs_P import *
from .go_funcs_Q import *
from .go_funcs_R import *
from .go_funcs_S import *
from .go_funcs_T import *
from .go_funcs_U import *
from .go_funcs_V import *
from .go_funcs_W import *
from .go_funcs_X import *
from .go_funcs_Y import *
from .go_funcs_Z import *
__all__ = [s for s in dir() if not s.startswith('_')]

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go_benchmark_functions/go_benchmark.py Normal file
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# -*- coding: utf-8 -*-
import numpy as np
from numpy import abs, asarray
from ..common import safe_import
with safe_import():
from scipy.special import factorial
class Benchmark:
"""
Defines a global optimization benchmark problem.
This abstract class defines the basic structure of a global
optimization problem. Subclasses should implement the ``fun`` method
for a particular optimization problem.
Attributes
----------
N : int
The dimensionality of the problem.
bounds : sequence
The lower/upper bounds to be used for minimizing the problem.
This a list of (lower, upper) tuples that contain the lower and upper
bounds for the problem. The problem should not be asked for evaluation
outside these bounds. ``len(bounds) == N``.
xmin : sequence
The lower bounds for the problem
xmax : sequence
The upper bounds for the problem
fglob : float
The global minimum of the evaluated function.
global_optimum : sequence
A list of vectors that provide the locations of the global minimum.
Note that some problems have multiple global minima, not all of which
may be listed.
nfev : int
the number of function evaluations that the object has been asked to
calculate.
change_dimensionality : bool
Whether we can change the benchmark function `x` variable length (i.e.,
the dimensionality of the problem)
custom_bounds : sequence
a list of tuples that contain lower/upper bounds for use in plotting.
"""
def __init__(self, dimensions):
"""
Initialises the problem
Parameters
----------
dimensions : int
The dimensionality of the problem
"""
self._dimensions = dimensions
self.nfev = 0
self.fglob = np.nan
self.global_optimum = None
self.change_dimensionality = False
self.custom_bounds = None
def __str__(self):
return '{0} ({1} dimensions)'.format(self.__class__.__name__, self.N)
def __repr__(self):
return self.__class__.__name__
def initial_vector(self):
"""
Random initialisation for the benchmark problem.
Returns
-------
x : sequence
a vector of length ``N`` that contains random floating point
numbers that lie between the lower and upper bounds for a given
parameter.
"""
return asarray([np.random.uniform(l, u) for l, u in self.bounds])
def success(self, x, tol=1.e-5):
"""
Tests if a candidate solution at the global minimum.
The default test is
Parameters
----------
x : sequence
The candidate vector for testing if the global minimum has been
reached. Must have ``len(x) == self.N``
tol : float
The evaluated function and known global minimum must differ by less
than this amount to be at a global minimum.
Returns
-------
bool : is the candidate vector at the global minimum?
"""
val = self.fun(asarray(x))
if abs(val - self.fglob) < tol:
return True
# the solution should still be in bounds, otherwise immediate fail.
if np.any(x > np.asfarray(self.bounds)[:, 1]):
return False
if np.any(x < np.asfarray(self.bounds)[:, 0]):
return False
# you found a lower global minimum. This shouldn't happen.
if val < self.fglob:
raise ValueError("Found a lower global minimum",
x,
val,
self.fglob)
return False
def fun(self, x):
"""
Evaluation of the benchmark function.
Parameters
----------
x : sequence
The candidate vector for evaluating the benchmark problem. Must
have ``len(x) == self.N``.
Returns
-------
val : float
the evaluated benchmark function
"""
raise NotImplementedError
def change_dimensions(self, ndim):
"""
Changes the dimensionality of the benchmark problem
The dimensionality will only be changed if the problem is suitable
Parameters
----------
ndim : int
The new dimensionality for the problem.
"""
if self.change_dimensionality:
self._dimensions = ndim
else:
raise ValueError('dimensionality cannot be changed for this'
'problem')
@property
def bounds(self):
"""
The lower/upper bounds to be used for minimizing the problem.
This a list of (lower, upper) tuples that contain the lower and upper
bounds for the problem. The problem should not be asked for evaluation
outside these bounds. ``len(bounds) == N``.
"""
if self.change_dimensionality:
return [self._bounds[0]] * self.N
else:
return self._bounds
@property
def N(self):
"""
The dimensionality of the problem.
Returns
-------
N : int
The dimensionality of the problem
"""
return self._dimensions
@property
def xmin(self):
"""
The lower bounds for the problem
Returns
-------
xmin : sequence
The lower bounds for the problem
"""
return asarray([b[0] for b in self.bounds])
@property
def xmax(self):
"""
The upper bounds for the problem
Returns
-------
xmax : sequence
The upper bounds for the problem
"""
return asarray([b[1] for b in self.bounds])

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# -*- coding: utf-8 -*-
from numpy import abs, cos, exp, pi, prod, sin, sqrt, sum
from .go_benchmark import Benchmark
class Ackley01(Benchmark):
r"""
Ackley01 objective function.
The Ackley01 [1]_ global optimization problem is a multimodal minimization
problem defined as follows:
.. math::
f_{\text{Ackley01}}(x) = -20 e^{-0.2 \sqrt{\frac{1}{n} \sum_{i=1}^n
x_i^2}} - e^{\frac{1}{n} \sum_{i=1}^n \cos(2 \pi x_i)} + 20 + e
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-35, 35]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Adorio, E. MVF - "Multivariate Test Functions Library in C for
Unconstrained Global Optimization", 2005
TODO: the -0.2 factor in the exponent of the first term is given as
-0.02 in Jamil et al.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-35.0] * self.N, [35.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
u = sum(x ** 2)
v = sum(cos(2 * pi * x))
return (-20. * exp(-0.2 * sqrt(u / self.N))
- exp(v / self.N) + 20. + exp(1.))
class Ackley02(Benchmark):
r"""
Ackley02 objective function.
The Ackley02 [1]_ global optimization problem is a multimodal minimization
problem defined as follows:
.. math::
f_{\text{Ackley02}(x) = -200 e^{-0.02 \sqrt{x_1^2 + x_2^2}}
with :math:`x_i \in [-32, 32]` for :math:`i=1, 2`.
*Global optimum*: :math:`f(x) = -200` for :math:`x = [0, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-32.0] * self.N, [32.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = -200.
def fun(self, x, *args):
self.nfev += 1
return -200 * exp(-0.02 * sqrt(x[0] ** 2 + x[1] ** 2))
class Ackley03(Benchmark):
r"""
Ackley03 [1]_ objective function.
The Ackley03 global optimization problem is a multimodal minimization
problem defined as follows:
.. math::
f_{\text{Ackley03}}(x) = -200 e^{-0.02 \sqrt{x_1^2 + x_2^2}} +
5e^{\cos(3x_1) + \sin(3x_2)}
with :math:`x_i \in [-32, 32]` for :math:`i=1, 2`.
*Global optimum*: :math:`f(x) = -195.62902825923879` for :math:`x
= [-0.68255758, -0.36070859]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: I think the minus sign is missing in front of the first term in eqn3
in [1]_. This changes the global minimum
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-32.0] * self.N, [32.0] * self.N))
self.global_optimum = [[-0.68255758, -0.36070859]]
self.fglob = -195.62902825923879
def fun(self, x, *args):
self.nfev += 1
a = -200 * exp(-0.02 * sqrt(x[0] ** 2 + x[1] ** 2))
a += 5 * exp(cos(3 * x[0]) + sin(3 * x[1]))
return a
class Adjiman(Benchmark):
r"""
Adjiman objective function.
The Adjiman [1]_ global optimization problem is a multimodal minimization
problem defined as follows:
.. math::
f_{\text{Adjiman}}(x) = \cos(x_1)\sin(x_2) - \frac{x_1}{(x_2^2 + 1)}
with, :math:`x_1 \in [-1, 2]` and :math:`x_2 \in [-1, 1]`.
*Global optimum*: :math:`f(x) = -2.02181` for :math:`x = [2.0, 0.10578]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = ([-1.0, 2.0], [-1.0, 1.0])
self.global_optimum = [[2.0, 0.10578]]
self.fglob = -2.02180678
def fun(self, x, *args):
self.nfev += 1
return cos(x[0]) * sin(x[1]) - x[0] / (x[1] ** 2 + 1)
class Alpine01(Benchmark):
r"""
Alpine01 objective function.
The Alpine01 [1]_ global optimization problem is a multimodal minimization
problem defined as follows:
.. math::
f_{\text{Alpine01}}(x) = \sum_{i=1}^{n} \lvert {x_i \sin \left( x_i
\right) + 0.1 x_i} \rvert
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return sum(abs(x * sin(x) + 0.1 * x))
class Alpine02(Benchmark):
r"""
Alpine02 objective function.
The Alpine02 [1]_ global optimization problem is a multimodal minimization
problem defined as follows:
.. math::
f_{\text{Alpine02}(x) = \prod_{i=1}^{n} \sqrt{x_i} \sin(x_i)
Here, :math:`n` represents the number of dimensions and :math:`x_i \in [0,
10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = -6.1295` for :math:`x =
[7.91705268, 4.81584232]` for :math:`i = 1, 2`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: eqn 7 in [1]_ has the wrong global minimum value.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [10.0] * self.N))
self.global_optimum = [[7.91705268, 4.81584232]]
self.fglob = -6.12950
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return prod(sqrt(x) * sin(x))
class AMGM(Benchmark):
r"""
AMGM objective function.
The AMGM (Arithmetic Mean - Geometric Mean Equality) global optimization
problem is a multimodal minimization problem defined as follows
.. math::
f_{\text{AMGM}}(x) = \left ( \frac{1}{n} \sum_{i=1}^{n} x_i -
\sqrt[n]{ \prod_{i=1}^{n} x_i} \right )^2
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[0, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_1 = x_2 = ... = x_n` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO, retrieved 2015
TODO: eqn 7 in [1]_ has the wrong global minimum value.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [10.0] * self.N))
self.global_optimum = [[1, 1]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
f1 = sum(x)
f2 = prod(x)
f1 = f1 / self.N
f2 = f2 ** (1.0 / self.N)
f = (f1 - f2) ** 2
return f

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# -*- coding: utf-8 -*-
from numpy import abs, cos, exp, log, arange, pi, sin, sqrt, sum
from .go_benchmark import Benchmark
class BartelsConn(Benchmark):
r"""
Bartels-Conn objective function.
The BartelsConn [1]_ global optimization problem is a multimodal
minimization problem defined as follows:
.. math::
f_{\text{BartelsConn}}(x) = \lvert {x_1^2 + x_2^2 + x_1x_2} \rvert +
\lvert {\sin(x_1)} \rvert + \lvert {\cos(x_2)} \rvert
with :math:`x_i \in [-500, 500]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 1` for :math:`x = [0, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-500.] * self.N, [500.] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 1.0
def fun(self, x, *args):
self.nfev += 1
return (abs(x[0] ** 2.0 + x[1] ** 2.0 + x[0] * x[1]) + abs(sin(x[0]))
+ abs(cos(x[1])))
class Beale(Benchmark):
r"""
Beale objective function.
The Beale [1]_ global optimization problem is a multimodal
minimization problem defined as follows:
.. math::
f_{\text{Beale}}(x) = \left(x_1 x_2 - x_1 + 1.5\right)^{2} +
\left(x_1 x_2^{2} - x_1 + 2.25\right)^{2} + \left(x_1 x_2^{3} - x_1 +
2.625\right)^{2}
with :math:`x_i \in [-4.5, 4.5]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x=[3, 0.5]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-4.5] * self.N, [4.5] * self.N))
self.global_optimum = [[3.0, 0.5]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return ((1.5 - x[0] + x[0] * x[1]) ** 2
+ (2.25 - x[0] + x[0] * x[1] ** 2) ** 2
+ (2.625 - x[0] + x[0] * x[1] ** 3) ** 2)
class BiggsExp02(Benchmark):
r"""
BiggsExp02 objective function.
The BiggsExp02 [1]_ global optimization problem is a multimodal minimization
problem defined as follows
.. math::
\begin{matrix}
f_{\text{BiggsExp02}}(x) = \sum_{i=1}^{10} (e^{-t_i x_1}
- 5 e^{-t_i x_2} - y_i)^2 \\
t_i = 0.1 i\\
y_i = e^{-t_i} - 5 e^{-10t_i}\\
\end{matrix}
with :math:`x_i \in [0, 20]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 10]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0] * 2,
[20] * 2))
self.global_optimum = [[1., 10.]]
self.fglob = 0
def fun(self, x, *args):
self.nfev += 1
t = arange(1, 11.) * 0.1
y = exp(-t) - 5 * exp(-10 * t)
vec = (exp(-t * x[0]) - 5 * exp(-t * x[1]) - y) ** 2
return sum(vec)
class BiggsExp03(Benchmark):
r"""
BiggsExp03 objective function.
The BiggsExp03 [1]_ global optimization problem is a multimodal minimization
problem defined as follows
.. math::
\begin{matrix}\ f_{\text{BiggsExp03}}(x) = \sum_{i=1}^{10}
(e^{-t_i x_1} - x_3e^{-t_i x_2} - y_i)^2\\
t_i = 0.1i\\
y_i = e^{-t_i} - 5e^{-10 t_i}\\
\end{matrix}
with :math:`x_i \in [0, 20]` for :math:`i = 1, 2, 3`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 10, 5]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=3):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0] * 3,
[20] * 3))
self.global_optimum = [[1., 10., 5.]]
self.fglob = 0
def fun(self, x, *args):
self.nfev += 1
t = arange(1., 11.) * 0.1
y = exp(-t) - 5 * exp(-10 * t)
vec = (exp(-t * x[0]) - x[2] * exp(-t * x[1]) - y) ** 2
return sum(vec)
class BiggsExp04(Benchmark):
r"""
BiggsExp04 objective function.
The BiggsExp04 [1]_ global optimization problem is a multimodal
minimization problem defined as follows
.. math::
\begin{matrix}\ f_{\text{BiggsExp04}}(x) = \sum_{i=1}^{10}
(x_3 e^{-t_i x_1} - x_4 e^{-t_i x_2} - y_i)^2\\
t_i = 0.1i\\
y_i = e^{-t_i} - 5 e^{-10 t_i}\\
\end{matrix}
with :math:`x_i \in [0, 20]` for :math:`i = 1, ..., 4`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 10, 1, 5]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=4):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.] * 4,
[20.] * 4))
self.global_optimum = [[1., 10., 1., 5.]]
self.fglob = 0
def fun(self, x, *args):
self.nfev += 1
t = arange(1, 11.) * 0.1
y = exp(-t) - 5 * exp(-10 * t)
vec = (x[2] * exp(-t * x[0]) - x[3] * exp(-t * x[1]) - y) ** 2
return sum(vec)
class BiggsExp05(Benchmark):
r"""
BiggsExp05 objective function.
The BiggsExp05 [1]_ global optimization problem is a multimodal minimization
problem defined as follows
.. math::
\begin{matrix}\ f_{\text{BiggsExp05}}(x) = \sum_{i=1}^{11}
(x_3 e^{-t_i x_1} - x_4 e^{-t_i x_2} + 3 e^{-t_i x_5} - y_i)^2\\
t_i = 0.1i\\
y_i = e^{-t_i} - 5e^{-10 t_i} + 3e^{-4 t_i}\\
\end{matrix}
with :math:`x_i \in [0, 20]` for :math:`i=1, ..., 5`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 10, 1, 5, 4]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=5):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.] * 5,
[20.] * 5))
self.global_optimum = [[1., 10., 1., 5., 4.]]
self.fglob = 0
def fun(self, x, *args):
self.nfev += 1
t = arange(1, 12.) * 0.1
y = exp(-t) - 5 * exp(-10 * t) + 3 * exp(-4 * t)
vec = (x[2] * exp(-t * x[0]) - x[3] * exp(-t * x[1])
+ 3 * exp(-t * x[4]) - y) ** 2
return sum(vec)
class Bird(Benchmark):
r"""
Bird objective function.
The Bird global optimization problem is a multimodal minimization
problem defined as follows
.. math::
f_{\text{Bird}}(x) = \left(x_1 - x_2\right)^{2} + e^{\left[1 -
\sin\left(x_1\right) \right]^{2}} \cos\left(x_2\right) + e^{\left[1 -
\cos\left(x_2\right)\right]^{2}} \sin\left(x_1\right)
with :math:`x_i \in [-2\pi, 2\pi]`
*Global optimum*: :math:`f(x) = -106.7645367198034` for :math:`x
= [4.701055751981055, 3.152946019601391]` or :math:`x =
[-1.582142172055011, -3.130246799635430]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-2.0 * pi] * self.N,
[2.0 * pi] * self.N))
self.global_optimum = [[4.701055751981055, 3.152946019601391],
[-1.582142172055011, -3.130246799635430]]
self.fglob = -106.7645367198034
def fun(self, x, *args):
self.nfev += 1
return (sin(x[0]) * exp((1 - cos(x[1])) ** 2)
+ cos(x[1]) * exp((1 - sin(x[0])) ** 2) + (x[0] - x[1]) ** 2)
class Bohachevsky1(Benchmark):
r"""
Bohachevsky 1 objective function.
The Bohachevsky 1 [1]_ global optimization problem is a multimodal
minimization problem defined as follows
.. math::
f_{\text{Bohachevsky}}(x) = \sum_{i=1}^{n-1}\left[x_i^2 + 2 x_{i+1}^2 -
0.3 \cos(3 \pi x_i) - 0.4 \cos(4 \pi x_{i + 1}) + 0.7 \right]
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-15, 15]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for :math:`i = 1,
..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: equation needs to be fixed up in the docstring. see Jamil#17
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-100.0] * self.N, [100.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return (x[0] ** 2 + 2 * x[1] ** 2 - 0.3 * cos(3 * pi * x[0])
- 0.4 * cos(4 * pi * x[1]) + 0.7)
class Bohachevsky2(Benchmark):
r"""
Bohachevsky 2 objective function.
The Bohachevsky 2 [1]_ global optimization problem is a multimodal
minimization problem defined as follows
.. math::
f_{\text{Bohachevsky}}(x) = \sum_{i=1}^{n-1}\left[x_i^2 + 2 x_{i+1}^2 -
0.3 \cos(3 \pi x_i) - 0.4 \cos(4 \pi x_{i + 1}) + 0.7 \right]
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-15, 15]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for :math:`i = 1,
..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: equation needs to be fixed up in the docstring. Jamil is also wrong.
There should be no 0.4 factor in front of the cos term
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-100.0] * self.N, [100.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return (x[0] ** 2 + 2 * x[1] ** 2 - 0.3 * cos(3 * pi * x[0])
* cos(4 * pi * x[1]) + 0.3)
class Bohachevsky3(Benchmark):
r"""
Bohachevsky 3 objective function.
The Bohachevsky 3 [1]_ global optimization problem is a multimodal
minimization problem defined as follows
.. math::
f_{\text{Bohachevsky}}(x) = \sum_{i=1}^{n-1}\left[x_i^2 + 2 x_{i+1}^2 -
0.3 \cos(3 \pi x_i) - 0.4 \cos(4 \pi x_{i + 1}) + 0.7 \right]
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-15, 15]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for :math:`i = 1,
..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: equation needs to be fixed up in the docstring. Jamil#19
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-100.0] * self.N, [100.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return (x[0] ** 2 + 2 * x[1] ** 2
- 0.3 * cos(3 * pi * x[0] + 4 * pi * x[1]) + 0.3)
class BoxBetts(Benchmark):
r"""
BoxBetts objective function.
The BoxBetts global optimization problem is a multimodal
minimization problem defined as follows
.. math::
f_{\text{BoxBetts}}(x) = \sum_{i=1}^k g(x_i)^2
Where, in this exercise:
.. math::
g(x) = e^{-0.1i x_1} - e^{-0.1i x_2} - x_3\left[e^{-0.1i}
- e^{-i}\right]
And :math:`k = 10`.
Here, :math:`x_1 \in [0.9, 1.2], x_2 \in [9, 11.2], x_3 \in [0.9, 1.2]`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 10, 1]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=3):
Benchmark.__init__(self, dimensions)
self._bounds = ([0.9, 1.2], [9.0, 11.2], [0.9, 1.2])
self.global_optimum = [[1.0, 10.0, 1.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
i = arange(1, 11)
g = (exp(-0.1 * i * x[0]) - exp(-0.1 * i * x[1])
- (exp(-0.1 * i) - exp(-i)) * x[2])
return sum(g**2)
class Branin01(Benchmark):
r"""
Branin01 objective function.
The Branin01 global optimization problem is a multimodal minimization
problem defined as follows
.. math::
f_{\text{Branin01}}(x) = \left(- 1.275 \frac{x_1^{2}}{\pi^{2}} + 5
\frac{x_1}{\pi} + x_2 -6\right)^{2} + \left(10 -\frac{5}{4 \pi} \right)
\cos\left(x_1\right) + 10
with :math:`x_1 \in [-5, 10], x_2 \in [0, 15]`
*Global optimum*: :math:`f(x) = 0.39788735772973816` for :math:`x =
[-\pi, 12.275]` or :math:`x = [\pi, 2.275]` or :math:`x = [3\pi, 2.475]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: Jamil#22, one of the solutions is different
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = [(-5., 10.), (0., 15.)]
self.global_optimum = [[-pi, 12.275], [pi, 2.275], [3 * pi, 2.475]]
self.fglob = 0.39788735772973816
def fun(self, x, *args):
self.nfev += 1
return ((x[1] - (5.1 / (4 * pi ** 2)) * x[0] ** 2
+ 5 * x[0] / pi - 6) ** 2
+ 10 * (1 - 1 / (8 * pi)) * cos(x[0]) + 10)
class Branin02(Benchmark):
r"""
Branin02 objective function.
The Branin02 global optimization problem is a multimodal minimization
problem defined as follows
.. math::
f_{\text{Branin02}}(x) = \left(- 1.275 \frac{x_1^{2}}{\pi^{2}}
+ 5 \frac{x_1}{\pi} + x_2 - 6 \right)^{2} + \left(10 - \frac{5}{4 \pi}
\right) \cos\left(x_1\right) \cos\left(x_2\right)
+ \log(x_1^2+x_2^2 + 1) + 10
with :math:`x_i \in [-5, 15]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 5.559037` for :math:`x = [-3.2, 12.53]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = [(-5.0, 15.0), (-5.0, 15.0)]
self.global_optimum = [[-3.1969884, 12.52625787]]
self.fglob = 5.5589144038938247
def fun(self, x, *args):
self.nfev += 1
return ((x[1] - (5.1 / (4 * pi ** 2)) * x[0] ** 2
+ 5 * x[0] / pi - 6) ** 2
+ 10 * (1 - 1 / (8 * pi)) * cos(x[0]) * cos(x[1])
+ log(x[0] ** 2.0 + x[1] ** 2.0 + 1.0) + 10)
class Brent(Benchmark):
r"""
Brent objective function.
The Brent [1]_ global optimization problem is a multimodal minimization
problem defined as follows:
.. math::
f_{\text{Brent}}(x) = (x_1 + 10)^2 + (x_2 + 10)^2 + e^{(-x_1^2 -x_2^2)}
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [-10, -10]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO solution is different to Jamil#24
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = ([-10, 2], [-10, 2])
self.global_optimum = [[-10.0, -10.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return ((x[0] + 10.0) ** 2.0 + (x[1] + 10.0) ** 2.0
+ exp(-x[0] ** 2.0 - x[1] ** 2.0))
class Brown(Benchmark):
r"""
Brown objective function.
The Brown [1]_ global optimization problem is a multimodal minimization
problem defined as follows:
.. math::
f_{\text{Brown}}(x) = \sum_{i=1}^{n-1}\left[
\left(x_i^2\right)^{x_{i + 1}^2 + 1}
+ \left(x_{i + 1}^2\right)^{x_i^2 + 1}\right]
with :math:`x_i \in [-1, 4]` for :math:`i=1,...,n`.
*Global optimum*: :math:`f(x_i) = 0` for :math:`x_i = 0` for
:math:`i=1,...,n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-1.0] * self.N, [4.0] * self.N))
self.custom_bounds = ([-1.0, 1.0], [-1.0, 1.0])
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
x0 = x[:-1]
x1 = x[1:]
return sum((x0 ** 2.0) ** (x1 ** 2.0 + 1.0)
+ (x1 ** 2.0) ** (x0 ** 2.0 + 1.0))
class Bukin02(Benchmark):
r"""
Bukin02 objective function.
The Bukin02 [1]_ global optimization problem is a multimodal minimization
problem defined as follows:
.. math::
f_{\text{Bukin02}}(x) = 100 (x_2^2 - 0.01x_1^2 + 1)
+ 0.01(x_1 + 10)^2
with :math:`x_1 \in [-15, -5], x_2 \in [-3, 3]`
*Global optimum*: :math:`f(x) = -124.75` for :math:`x = [-15, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: I think that Gavana and Jamil are wrong on this function. In both
sources the x[1] term is not squared. As such there will be a minimum at
the smallest value of x[1].
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = [(-15.0, -5.0), (-3.0, 3.0)]
self.global_optimum = [[-15.0, 0.0]]
self.fglob = -124.75
def fun(self, x, *args):
self.nfev += 1
return (100 * (x[1] ** 2 - 0.01 * x[0] ** 2 + 1.0)
+ 0.01 * (x[0] + 10.0) ** 2.0)
class Bukin04(Benchmark):
r"""
Bukin04 objective function.
The Bukin04 [1]_ global optimization problem is a multimodal minimization
problem defined as follows:
.. math::
f_{\text{Bukin04}}(x) = 100 x_2^{2} + 0.01 \lvert{x_1 + 10}
\rvert
with :math:`x_1 \in [-15, -5], x_2 \in [-3, 3]`
*Global optimum*: :math:`f(x) = 0` for :math:`x = [-10, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = [(-15.0, -5.0), (-3.0, 3.0)]
self.global_optimum = [[-10.0, 0.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return 100 * x[1] ** 2 + 0.01 * abs(x[0] + 10)
class Bukin06(Benchmark):
r"""
Bukin06 objective function.
The Bukin06 [1]_ global optimization problem is a multimodal minimization
problem defined as follows:
.. math::
f_{\text{Bukin06}}(x) = 100 \sqrt{ \lvert{x_2 - 0.01 x_1^{2}}
\rvert} + 0.01 \lvert{x_1 + 10} \rvert
with :math:`x_1 \in [-15, -5], x_2 \in [-3, 3]`
*Global optimum*: :math:`f(x) = 0` for :math:`x = [-10, 1]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = [(-15.0, -5.0), (-3.0, 3.0)]
self.global_optimum = [[-10.0, 1.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return 100 * sqrt(abs(x[1] - 0.01 * x[0] ** 2)) + 0.01 * abs(x[0] + 10)

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go_benchmark_functions/go_funcs_C.py Normal file
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# -*- coding: utf-8 -*-
import numpy as np
from numpy import (abs, asarray, cos, exp, floor, pi, sign, sin, sqrt, sum,
size, tril, isnan, atleast_2d, repeat)
from numpy.testing import assert_almost_equal
from .go_benchmark import Benchmark
class CarromTable(Benchmark):
r"""
CarromTable objective function.
The CarromTable [1]_ global optimization problem is a multimodal
minimization problem defined as follows:
.. math::
f_{\text{CarromTable}}(x) = - \frac{1}{30}\left(\cos(x_1)
cos(x_2) e^{\left|1 - \frac{\sqrt{x_1^2 + x_2^2}}{\pi}\right|}\right)^2
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -24.15681551650653` for :math:`x_i = \pm
9.646157266348881` for :math:`i = 1, 2`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [(9.646157266348881, 9.646134286497169),
(-9.646157266348881, 9.646134286497169),
(9.646157266348881, -9.646134286497169),
(-9.646157266348881, -9.646134286497169)]
self.fglob = -24.15681551650653
def fun(self, x, *args):
self.nfev += 1
u = cos(x[0]) * cos(x[1])
v = sqrt(x[0] ** 2 + x[1] ** 2)
return -((u * exp(abs(1 - v / pi))) ** 2) / 30.
class Chichinadze(Benchmark):
r"""
Chichinadze objective function.
This class defines the Chichinadze [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Chichinadze}}(x) = x_{1}^{2} - 12 x_{1}
+ 8 \sin\left(\frac{5}{2} \pi x_{1}\right)
+ 10 \cos\left(\frac{1}{2} \pi x_{1}\right) + 11
- 0.2 \frac{\sqrt{5}}{e^{\frac{1}{2} \left(x_{2} -0.5 \right)^{2}}}
with :math:`x_i \in [-30, 30]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -42.94438701899098` for :math:`x =
[6.189866586965680, 0.5]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO: Jamil#33 has a dividing factor of 2 in the sin term. However, f(x)
for the given solution does not give the global minimum. i.e. the equation
is at odds with the solution.
Only by removing the dividing factor of 2, i.e. `8 * sin(5 * pi * x[0])`
does the given solution result in the given global minimum.
Do we keep the result or equation?
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-30.0] * self.N, [30.0] * self.N))
self.custom_bounds = [(-10, 10), (-10, 10)]
self.global_optimum = [[6.189866586965680, 0.5]]
self.fglob = -42.94438701899098
def fun(self, x, *args):
self.nfev += 1
return (x[0] ** 2 - 12 * x[0] + 11 + 10 * cos(pi * x[0] / 2)
+ 8 * sin(5 * pi * x[0] / 2)
- 1.0 / sqrt(5) * exp(-((x[1] - 0.5) ** 2) / 2))
class Cigar(Benchmark):
r"""
Cigar objective function.
This class defines the Cigar [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Cigar}}(x) = x_1^2 + 10^6\sum_{i=2}^{n} x_i^2
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-100, 100]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-100.0] * self.N,
[100.0] * self.N))
self.custom_bounds = [(-5, 5), (-5, 5)]
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return x[0] ** 2 + 1e6 * sum(x[1:] ** 2)
class Cola(Benchmark):
r"""
Cola objective function.
This class defines the Cola global optimization problem. The 17-dimensional
function computes indirectly the formula :math:`f(n, u)` by setting
:math:`x_0 = y_0, x_1 = u_0, x_i = u_{2(i2)}, y_i = u_{2(i2)+1}` :
.. math::
f_{\text{Cola}}(x) = \sum_{i<j}^{n} \left (r_{i,j} - d_{i,j} \right )^2
Where :math:`r_{i, j}` is given by:
.. math::
r_{i, j} = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}
And :math:`d` is a symmetric matrix given by:
.. math::
\{d} = \left [ d_{ij} \right ] = \begin{pmatrix}
1.27 & & & & & & & & \\
1.69 & 1.43 & & & & & & & \\
2.04 & 2.35 & 2.43 & & & & & & \\
3.09 & 3.18 & 3.26 & 2.85 & & & & & \\
3.20 & 3.22 & 3.27 & 2.88 & 1.55 & & & & \\
2.86 & 2.56 & 2.58 & 2.59 & 3.12 & 3.06 & & & \\
3.17 & 3.18 & 3.18 & 3.12 & 1.31 & 1.64 & 3.00 & \\
3.21 & 3.18 & 3.18 & 3.17 & 1.70 & 1.36 & 2.95 & 1.32 & \\
2.38 & 2.31 & 2.42 & 1.94 & 2.85 & 2.81 & 2.56 & 2.91 & 2.97
\end{pmatrix}
This function has bounds :math:`x_0 \in [0, 4]` and :math:`x_i \in [-4, 4]`
for :math:`i = 1, ..., n-1`.
*Global optimum* 11.7464.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=17):
Benchmark.__init__(self, dimensions)
self._bounds = [[0.0, 4.0]] + list(zip([-4.0] * (self.N - 1),
[4.0] * (self.N - 1)))
self.global_optimum = [[0.651906, 1.30194, 0.099242, -0.883791,
-0.8796, 0.204651, -3.28414, 0.851188,
-3.46245, 2.53245, -0.895246, 1.40992,
-3.07367, 1.96257, -2.97872, -0.807849,
-1.68978]]
self.fglob = 11.7464
self.d = asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[1.27, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[1.69, 1.43, 0, 0, 0, 0, 0, 0, 0, 0],
[2.04, 2.35, 2.43, 0, 0, 0, 0, 0, 0, 0],
[3.09, 3.18, 3.26, 2.85, 0, 0, 0, 0, 0, 0],
[3.20, 3.22, 3.27, 2.88, 1.55, 0, 0, 0, 0, 0],
[2.86, 2.56, 2.58, 2.59, 3.12, 3.06, 0, 0, 0, 0],
[3.17, 3.18, 3.18, 3.12, 1.31, 1.64, 3.00, 0, 0, 0],
[3.21, 3.18, 3.18, 3.17, 1.70, 1.36, 2.95, 1.32, 0, 0],
[2.38, 2.31, 2.42, 1.94, 2.85, 2.81, 2.56, 2.91, 2.97, 0.]])
def fun(self, x, *args):
self.nfev += 1
xi = atleast_2d(asarray([0.0, x[0]] + list(x[1::2])))
xj = repeat(xi, size(xi, 1), axis=0)
xi = xi.T
yi = atleast_2d(asarray([0.0, 0.0] + list(x[2::2])))
yj = repeat(yi, size(yi, 1), axis=0)
yi = yi.T
inner = (sqrt(((xi - xj) ** 2 + (yi - yj) ** 2)) - self.d) ** 2
inner = tril(inner, -1)
return sum(sum(inner, axis=1))
class Colville(Benchmark):
r"""
Colville objective function.
This class defines the Colville global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Colville}}(x) = \left(x_{1} -1\right)^{2}
+ 100 \left(x_{1}^{2} - x_{2}\right)^{2}
+ 10.1 \left(x_{2} -1\right)^{2} + \left(x_{3} -1\right)^{2}
+ 90 \left(x_{3}^{2} - x_{4}\right)^{2}
+ 10.1 \left(x_{4} -1\right)^{2} + 19.8 \frac{x_{4} -1}{x_{2}}
with :math:`x_i \in [-10, 10]` for :math:`i = 1, ..., 4`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 1` for
:math:`i = 1, ..., 4`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO docstring equation is wrong use Jamil#36
"""
def __init__(self, dimensions=4):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[1 for _ in range(self.N)]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return (100 * (x[0] - x[1] ** 2) ** 2
+ (1 - x[0]) ** 2 + (1 - x[2]) ** 2
+ 90 * (x[3] - x[2] ** 2) ** 2
+ 10.1 * ((x[1] - 1) ** 2 + (x[3] - 1) ** 2)
+ 19.8 * (x[1] - 1) * (x[3] - 1))
class Corana(Benchmark):
r"""
Corana objective function.
This class defines the Corana [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Corana}}(x) = \begin{cases} \sum_{i=1}^n 0.15 d_i
[z_i - 0.05\textrm{sgn}(z_i)]^2 & \textrm{if }|x_i-z_i| < 0.05 \\
d_ix_i^2 & \textrm{otherwise}\end{cases}
Where, in this exercise:
.. math::
z_i = 0.2 \lfloor |x_i/s_i|+0.49999\rfloor\textrm{sgn}(x_i),
d_i=(1,1000,10,100, ...)
with :math:`x_i \in [-5, 5]` for :math:`i = 1, ..., 4`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., 4`
..[1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=4):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
d = [1., 1000., 10., 100.]
r = 0
for j in range(4):
zj = floor(abs(x[j] / 0.2) + 0.49999) * sign(x[j]) * 0.2
if abs(x[j] - zj) < 0.05:
r += 0.15 * ((zj - 0.05 * sign(zj)) ** 2) * d[j]
else:
r += d[j] * x[j] * x[j]
return r
class CosineMixture(Benchmark):
r"""
Cosine Mixture objective function.
This class defines the Cosine Mixture global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{CosineMixture}}(x) = -0.1 \sum_{i=1}^n \cos(5 \pi x_i)
- \sum_{i=1}^n x_i^2
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-1, 1]` for :math:`i = 1, ..., N`.
*Global optimum*: :math:`f(x) = -0.1N` for :math:`x_i = 0` for
:math:`i = 1, ..., N`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO, Jamil #38 has wrong minimum and wrong fglob. I plotted it.
-(x**2) term is always negative if x is negative.
cos(5 * pi * x) is equal to -1 for x=-1.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self.change_dimensionality = True
self._bounds = list(zip([-1.0] * self.N, [1.0] * self.N))
self.global_optimum = [[-1. for _ in range(self.N)]]
self.fglob = -0.9 * self.N
def fun(self, x, *args):
self.nfev += 1
return -0.1 * sum(cos(5.0 * pi * x)) - sum(x ** 2.0)
class CrossInTray(Benchmark):
r"""
Cross-in-Tray objective function.
This class defines the Cross-in-Tray [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{CrossInTray}}(x) = - 0.0001 \left(\left|{e^{\left|{100
- \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{\pi}}\right|}
\sin\left(x_{1}\right) \sin\left(x_{2}\right)}\right| + 1\right)^{0.1}
with :math:`x_i \in [-15, 15]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -2.062611870822739` for :math:`x_i =
\pm 1.349406608602084` for :math:`i = 1, 2`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [(1.349406685353340, 1.349406608602084),
(-1.349406685353340, 1.349406608602084),
(1.349406685353340, -1.349406608602084),
(-1.349406685353340, -1.349406608602084)]
self.fglob = -2.062611870822739
def fun(self, x, *args):
self.nfev += 1
return (-0.0001 * (abs(sin(x[0]) * sin(x[1])
* exp(abs(100 - sqrt(x[0] ** 2 + x[1] ** 2) / pi)))
+ 1) ** (0.1))
class CrossLegTable(Benchmark):
r"""
Cross-Leg-Table objective function.
This class defines the Cross-Leg-Table [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{CrossLegTable}}(x) = - \frac{1}{\left(\left|{e^{\left|{100
- \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{\pi}}\right|}
\sin\left(x_{1}\right) \sin\left(x_{2}\right)}\right| + 1\right)^{0.1}}
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -1`. The global minimum is found on the
planes :math:`x_1 = 0` and :math:`x_2 = 0`
..[1] Mishra, S. Global Optimization by Differential Evolution and Particle
Swarm Methods: Evaluation on Some Benchmark Functions Munich University,
2006
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[0., 0.]]
self.fglob = -1.0
def fun(self, x, *args):
self.nfev += 1
u = 100 - sqrt(x[0] ** 2 + x[1] ** 2) / pi
v = sin(x[0]) * sin(x[1])
return -(abs(v * exp(abs(u))) + 1) ** (-0.1)
class CrownedCross(Benchmark):
r"""
Crowned Cross objective function.
This class defines the Crowned Cross [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{CrownedCross}}(x) = 0.0001 \left(\left|{e^{\left|{100
- \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{\pi}}\right|}
\sin\left(x_{1}\right) \sin\left(x_{2}\right)}\right| + 1\right)^{0.1}
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x_i) = 0.0001`. The global minimum is found on
the planes :math:`x_1 = 0` and :math:`x_2 = 0`
..[1] Mishra, S. Global Optimization by Differential Evolution and Particle
Swarm Methods: Evaluation on Some Benchmark Functions Munich University,
2006
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[0, 0]]
self.fglob = 0.0001
def fun(self, x, *args):
self.nfev += 1
u = 100 - sqrt(x[0] ** 2 + x[1] ** 2) / pi
v = sin(x[0]) * sin(x[1])
return 0.0001 * (abs(v * exp(abs(u))) + 1) ** (0.1)
class Csendes(Benchmark):
r"""
Csendes objective function.
This class defines the Csendes [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Csendes}}(x) = \sum_{i=1}^n x_i^6 \left[ 2 + \sin
\left( \frac{1}{x_i} \right ) \right]
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-1, 1]` for :math:`i = 1, ..., N`.
*Global optimum*: :math:`f(x) = 0.0` for :math:`x_i = 0` for
:math:`i = 1, ..., N`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self.change_dimensionality = True
self._bounds = list(zip([-1.0] * self.N, [1.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = np.nan
def fun(self, x, *args):
self.nfev += 1
try:
return sum((x ** 6.0) * (2.0 + sin(1.0 / x)))
except ZeroDivisionError:
return np.nan
except FloatingPointError:
return np.nan
def success(self, x):
"""Is a candidate solution at the global minimum"""
val = self.fun(asarray(x))
if isnan(val):
return True
try:
assert_almost_equal(val, 0., 4)
return True
except AssertionError:
return False
return False
class Cube(Benchmark):
r"""
Cube objective function.
This class defines the Cube global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Cube}}(x) = 100(x_2 - x_1^3)^2 + (1 - x1)^2
Here, :math:`n` represents the number of dimensions and :math:`x_i \in [-10, 10]` for :math:`i=1,...,N`.
*Global optimum*: :math:`f(x_i) = 0.0` for :math:`x = [1, 1]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: jamil#41 has the wrong solution.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = ([0, 2], [0, 2])
self.global_optimum = [[1.0, 1.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return 100.0 * (x[1] - x[0] ** 3.0) ** 2.0 + (1.0 - x[0]) ** 2.0

568
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# -*- coding: utf-8 -*-
import numpy as np
from numpy import abs, cos, exp, arange, pi, sin, sqrt, sum, zeros, tanh
from numpy.testing import assert_almost_equal
from .go_benchmark import Benchmark
class Damavandi(Benchmark):
r"""
Damavandi objective function.
This class defines the Damavandi [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Damavandi}}(x) = \left[ 1 - \lvert{\frac{
\sin[\pi (x_1 - 2)]\sin[\pi (x2 - 2)]}{\pi^2 (x_1 - 2)(x_2 - 2)}}
\rvert^5 \right] \left[2 + (x_1 - 7)^2 + 2(x_2 - 7)^2 \right]
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[0, 14]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0.0` for :math:`x_i = 2` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, 2)
self._bounds = list(zip([0.0] * self.N, [14.0] * self.N))
self.global_optimum = [[2 for _ in range(self.N)]]
self.fglob = np.nan
def fun(self, x, *args):
self.nfev += 1
try:
num = sin(pi * (x[0] - 2.0)) * sin(pi * (x[1] - 2.0))
den = (pi ** 2) * (x[0] - 2.0) * (x[1] - 2.0)
factor1 = 1.0 - (abs(num / den)) ** 5.0
factor2 = 2 + (x[0] - 7.0) ** 2.0 + 2 * (x[1] - 7.0) ** 2.0
return factor1 * factor2
except ZeroDivisionError:
return np.nan
def success(self, x):
"""Is a candidate solution at the global minimum"""
val = self.fun(x)
if np.isnan(val):
return True
try:
assert_almost_equal(val, 0., 4)
return True
except AssertionError:
return False
return False
class Deb01(Benchmark):
r"""
Deb 1 objective function.
This class defines the Deb 1 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Deb01}}(x) = - \frac{1}{N} \sum_{i=1}^n \sin^6(5 \pi x_i)
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-1, 1]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x_i) = 0.0`. The number of global minima is
:math:`5^n` that are evenly spaced in the function landscape, where
:math:`n` represents the dimension of the problem.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self.change_dimensionality = True
self._bounds = list(zip([-1.0] * self.N, [1.0] * self.N))
self.global_optimum = [[0.3, -0.3]]
self.fglob = -1.0
def fun(self, x, *args):
self.nfev += 1
return -(1.0 / self.N) * sum(sin(5 * pi * x) ** 6.0)
class Deb03(Benchmark):
r"""
Deb 3 objective function.
This class defines the Deb 3 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Deb02}}(x) = - \frac{1}{N} \sum_{i=1}^n \sin^6 \left[ 5 \pi
\left ( x_i^{3/4} - 0.05 \right) \right ]
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[0, 1]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0.0`. The number of global minima is
:math:`5^n` that are evenly spaced in the function landscape, where
:math:`n` represents the dimension of the problem.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self.change_dimensionality = True
self._bounds = list(zip([-1.0] * self.N, [1.0] * self.N))
self.global_optimum = [[0.93388314, 0.68141781]]
self.fglob = -1.0
def fun(self, x, *args):
self.nfev += 1
return -(1.0 / self.N) * sum(sin(5 * pi * (x ** 0.75 - 0.05)) ** 6.0)
class Decanomial(Benchmark):
r"""
Decanomial objective function.
This class defines the Decanomial function global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Decanomial}}(x) = 0.001 \left(\lvert{x_{2}^{4} + 12 x_{2}^{3}
+ 54 x_{2}^{2} + 108 x_{2} + 81.0}\rvert + \lvert{x_{1}^{10}
- 20 x_{1}^{9} + 180 x_{1}^{8} - 960 x_{1}^{7} + 3360 x_{1}^{6}
- 8064 x_{1}^{5} + 13340 x_{1}^{4} - 15360 x_{1}^{3} + 11520 x_{1}^{2}
- 5120 x_{1} + 2624.0}\rvert\right)^{2}
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [2, -3]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(0, 2.5), (-2, -4)]
self.global_optimum = [[2.0, -3.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
val = x[1] ** 4 + 12 * x[1] ** 3 + 54 * x[1] ** 2 + 108 * x[1] + 81.0
val2 = x[0] ** 10. - 20 * x[0] ** 9 + 180 * x[0] ** 8 - 960 * x[0] ** 7
val2 += 3360 * x[0] ** 6 - 8064 * x[0] ** 5 + 13340 * x[0] ** 4
val2 += - 15360 * x[0] ** 3 + 11520 * x[0] ** 2 - 5120 * x[0] + 2624
return 0.001 * (abs(val) + abs(val2)) ** 2.
class Deceptive(Benchmark):
r"""
Deceptive objective function.
This class defines the Deceptive [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Deceptive}}(x) = - \left [\frac{1}{n}
\sum_{i=1}^{n} g_i(x_i) \right ]^{\beta}
Where :math:`\beta` is a fixed non-linearity factor; in this exercise,
:math:`\beta = 2`. The function :math:`g_i(x_i)` is given by:
.. math::
g_i(x_i) = \begin{cases}
- \frac{x}{\alpha_i} + \frac{4}{5} &
\textrm{if} \hspace{5pt} 0 \leq x_i \leq \frac{4}{5} \alpha_i \\
\frac{5x}{\alpha_i} -4 &
\textrm{if} \hspace{5pt} \frac{4}{5} \alpha_i \le x_i \leq \alpha_i \\
\frac{5(x - \alpha_i)}{\alpha_i-1} &
\textrm{if} \hspace{5pt} \alpha_i \le x_i \leq \frac{1 + 4\alpha_i}{5} \\
\frac{x - 1}{1 - \alpha_i} &
\textrm{if} \hspace{5pt} \frac{1 + 4\alpha_i}{5} \le x_i \leq 1
\end{cases}
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[0, 1]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = -1` for :math:`x_i = \alpha_i` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO: this function was taken from the Gavana website. The following code
is based on his code. His code and the website don't match, the equations
are wrong.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [1.0] * self.N))
alpha = arange(1.0, self.N + 1.0) / (self.N + 1.0)
self.global_optimum = [alpha]
self.fglob = -1.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
alpha = arange(1.0, self.N + 1.0) / (self.N + 1.0)
beta = 2.0
g = zeros((self.N, ))
for i in range(self.N):
if x[i] <= 0.0:
g[i] = x[i]
elif x[i] < 0.8 * alpha[i]:
g[i] = -x[i] / alpha[i] + 0.8
elif x[i] < alpha[i]:
g[i] = 5.0 * x[i] / alpha[i] - 4.0
elif x[i] < (1.0 + 4 * alpha[i]) / 5.0:
g[i] = 5.0 * (x[i] - alpha[i]) / (alpha[i] - 1.0) + 1.0
elif x[i] <= 1.0:
g[i] = (x[i] - 1.0) / (1.0 - alpha[i]) + 4.0 / 5.0
else:
g[i] = x[i] - 1.0
return -((1.0 / self.N) * sum(g)) ** beta
class DeckkersAarts(Benchmark):
r"""
Deckkers-Aarts objective function.
This class defines the Deckkers-Aarts [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{DeckkersAarts}}(x) = 10^5x_1^2 + x_2^2 - (x_1^2 + x_2^2)^2
+ 10^{-5}(x_1^2 + x_2^2)^4
with :math:`x_i \in [-20, 20]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -24776.518242168` for
:math:`x = [0, \pm 14.9451209]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: jamil solution and global minimum are slightly wrong.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-20.0] * self.N, [20.0] * self.N))
self.custom_bounds = ([-1, 1], [14, 16])
self.global_optimum = [[0.0, 14.9451209]]
self.fglob = -24776.518342168
def fun(self, x, *args):
self.nfev += 1
return (1.e5 * x[0] ** 2 + x[1] ** 2 - (x[0] ** 2 + x[1] ** 2) ** 2
+ 1.e-5 * (x[0] ** 2 + x[1] ** 2) ** 4)
class DeflectedCorrugatedSpring(Benchmark):
r"""
DeflectedCorrugatedSpring objective function.
This class defines the Deflected Corrugated Spring [1]_ function global
optimization problem. This is a multimodal minimization problem defined as
follows:
.. math::
f_{\text{DeflectedCorrugatedSpring}}(x) = 0.1\sum_{i=1}^n \left[ (x_i -
\alpha)^2 - \cos \left( K \sqrt {\sum_{i=1}^n (x_i - \alpha)^2}
\right ) \right ]
Where, in this exercise, :math:`K = 5` and :math:`\alpha = 5`.
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[0, 2\alpha]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = -1` for :math:`x_i = \alpha` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO: website has a different equation to the gavana codebase. The function
below is different to the equation above. Also, the global minimum is
wrong.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
alpha = 5.0
self._bounds = list(zip([0] * self.N, [2 * alpha] * self.N))
self.global_optimum = [[alpha for _ in range(self.N)]]
self.fglob = -1.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
K, alpha = 5.0, 5.0
return (-cos(K * sqrt(sum((x - alpha) ** 2)))
+ 0.1 * sum((x - alpha) ** 2))
class DeVilliersGlasser01(Benchmark):
r"""
DeVilliers-Glasser 1 objective function.
This class defines the DeVilliers-Glasser 1 [1]_ function global optimization
problem. This is a multimodal minimization problem defined as follows:
.. math::
f_{\text{DeVilliersGlasser01}}(x) = \sum_{i=1}^{24} \left[ x_1x_2^{t_i}
\sin(x_3t_i + x_4) - y_i \right ]^2
Where, in this exercise, :math:`t_i = 0.1(i - 1)` and
:math:`y_i = 60.137(1.371^{t_i}) \sin(3.112t_i + 1.761)`.
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[1, 100]` for :math:`i = 1, ..., 4`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`x = [60.137, 1.371, 3.112, 1.761]`.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=4):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([1.0] * self.N, [100.0] * self.N))
self.global_optimum = [[60.137, 1.371, 3.112, 1.761]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
t = 0.1 * arange(24)
y = 60.137 * (1.371 ** t) * sin(3.112 * t + 1.761)
return sum((x[0] * (x[1] ** t) * sin(x[2] * t + x[3]) - y) ** 2.0)
class DeVilliersGlasser02(Benchmark):
r"""
DeVilliers-Glasser 2 objective function.
This class defines the DeVilliers-Glasser 2 [1]_ function global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{DeVilliersGlasser01}}(x) = \sum_{i=1}^{24} \left[ x_1x_2^{t_i}
\tanh \left [x_3t_i + \sin(x_4t_i) \right] \cos(t_ie^{x_5}) -
y_i \right ]^2
Where, in this exercise, :math:`t_i = 0.1(i - 1)` and
:math:`y_i = 53.81(1.27^{t_i}) \tanh (3.012t_i + \sin(2.13t_i))
\cos(e^{0.507}t_i)`.
with :math:`x_i \in [1, 60]` for :math:`i = 1, ..., 5`.
*Global optimum*: :math:`f(x) = 0` for
:math:`x = [53.81, 1.27, 3.012, 2.13, 0.507]`.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=5):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([1.0] * self.N, [60.0] * self.N))
self.global_optimum = [[53.81, 1.27, 3.012, 2.13, 0.507]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
t = 0.1 * arange(16)
y = (53.81 * 1.27 ** t * tanh(3.012 * t + sin(2.13 * t))
* cos(exp(0.507) * t))
return sum((x[0] * (x[1] ** t) * tanh(x[2] * t + sin(x[3] * t))
* cos(t * exp(x[4])) - y) ** 2.0)
class DixonPrice(Benchmark):
r"""
Dixon and Price objective function.
This class defines the Dixon and Price global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{DixonPrice}}(x) = (x_i - 1)^2
+ \sum_{i=2}^n i(2x_i^2 - x_{i-1})^2
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x_i) = 0` for
:math:`x_i = 2^{- \frac{(2^i - 2)}{2^i}}` for :math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: Gavana code not correct. i array should start from 2.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(-2, 3), (-2, 3)]
self.global_optimum = [[2.0 ** (-(2.0 ** i - 2.0) / 2.0 ** i)
for i in range(1, self.N + 1)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
i = arange(2, self.N + 1)
s = i * (2.0 * x[1:] ** 2.0 - x[:-1]) ** 2.0
return sum(s) + (x[0] - 1.0) ** 2.0
class Dolan(Benchmark):
r"""
Dolan objective function.
This class defines the Dolan [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Dolan}}(x) = \lvert (x_1 + 1.7 x_2)\sin(x_1) - 1.5 x_3
- 0.1 x_4\cos(x_5 + x_5 - x_1) + 0.2 x_5^2 - x_2 - 1 \rvert
with :math:`x_i \in [-100, 100]` for :math:`i = 1, ..., 5`.
*Global optimum*: :math:`f(x_i) = 10^{-5}` for
:math:`x = [8.39045925, 4.81424707, 7.34574133, 68.88246895, 3.85470806]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO Jamil equation is missing the absolute brackets around the entire
expression.
"""
def __init__(self, dimensions=5):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-100.0] * self.N,
[100.0] * self.N))
self.global_optimum = [[-74.10522498, 44.33511286, 6.21069214,
18.42772233, -16.5839403]]
self.fglob = 0
def fun(self, x, *args):
self.nfev += 1
return (abs((x[0] + 1.7 * x[1]) * sin(x[0]) - 1.5 * x[2]
- 0.1 * x[3] * cos(x[3] + x[4] - x[0]) + 0.2 * x[4] ** 2
- x[1] - 1))
class DropWave(Benchmark):
r"""
DropWave objective function.
This class defines the DropWave [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{DropWave}}(x) = - \frac{1 + \cos\left(12 \sqrt{\sum_{i=1}^{n}
x_i^{2}}\right)}{2 + 0.5 \sum_{i=1}^{n} x_i^{2}}
with :math:`x_i \in [-5.12, 5.12]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -1` for :math:`x = [0, 0]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.12] * self.N, [5.12] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = -1.0
def fun(self, x, *args):
self.nfev += 1
norm_x = sum(x ** 2)
return -(1 + cos(12 * sqrt(norm_x))) / (0.5 * norm_x + 2)

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# -*- coding: utf-8 -*-
from numpy import abs, asarray, cos, exp, arange, pi, sin, sqrt, sum
from .go_benchmark import Benchmark
class Easom(Benchmark):
r"""
Easom objective function.
This class defines the Easom [1]_ global optimization problem. This is a
a multimodal minimization problem defined as follows:
.. math::
f_{\text{Easom}}({x}) = a - \frac{a}{e^{b \sqrt{\frac{\sum_{i=1}^{n}
x_i^{2}}{n}}}} + e - e^{\frac{\sum_{i=1}^{n} \cos\left(c x_i\right)}
{n}}
Where, in this exercise, :math:`a = 20, b = 0.2` and :math:`c = 2 \pi`.
Here, :math:`x_i \in [-100, 100]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO Gavana website disagrees with Jamil, etc. Gavana equation in docstring is totally wrong.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-100.0] * self.N,
[100.0] * self.N))
self.global_optimum = [[pi for _ in range(self.N)]]
self.fglob = -1.0
def fun(self, x, *args):
self.nfev += 1
a = (x[0] - pi)**2 + (x[1] - pi)**2
return -cos(x[0]) * cos(x[1]) * exp(-a)
class Eckerle4(Benchmark):
r"""
Eckerle4 objective function.
Eckerle, K., NIST (1979).
Circular Interference Transmittance Study.
..[1] https://www.itl.nist.gov/div898/strd/nls/data/eckerle4.shtml
#TODO, this is a NIST regression standard dataset, docstring needs
improving
"""
def __init__(self, dimensions=3):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0., 1., 10.],
[20, 20., 600.]))
self.global_optimum = [[1.5543827178, 4.0888321754, 4.5154121844e2]]
self.fglob = 1.4635887487E-03
self.a = asarray([1.5750000E-04, 1.6990000E-04, 2.3500000E-04,
3.1020000E-04, 4.9170000E-04, 8.7100000E-04,
1.7418000E-03, 4.6400000E-03, 6.5895000E-03,
9.7302000E-03, 1.4900200E-02, 2.3731000E-02,
4.0168300E-02, 7.1255900E-02, 1.2644580E-01,
2.0734130E-01, 2.9023660E-01, 3.4456230E-01,
3.6980490E-01, 3.6685340E-01, 3.1067270E-01,
2.0781540E-01, 1.1643540E-01, 6.1676400E-02,
3.3720000E-02, 1.9402300E-02, 1.1783100E-02,
7.4357000E-03, 2.2732000E-03, 8.8000000E-04,
4.5790000E-04, 2.3450000E-04, 1.5860000E-04,
1.1430000E-04, 7.1000000E-05])
self.b = asarray([4.0000000E+02, 4.0500000E+02, 4.1000000E+02,
4.1500000E+02, 4.2000000E+02, 4.2500000E+02,
4.3000000E+02, 4.3500000E+02, 4.3650000E+02,
4.3800000E+02, 4.3950000E+02, 4.4100000E+02,
4.4250000E+02, 4.4400000E+02, 4.4550000E+02,
4.4700000E+02, 4.4850000E+02, 4.5000000E+02,
4.5150000E+02, 4.5300000E+02, 4.5450000E+02,
4.5600000E+02, 4.5750000E+02, 4.5900000E+02,
4.6050000E+02, 4.6200000E+02, 4.6350000E+02,
4.6500000E+02, 4.7000000E+02, 4.7500000E+02,
4.8000000E+02, 4.8500000E+02, 4.9000000E+02,
4.9500000E+02, 5.0000000E+02])
def fun(self, x, *args):
self.nfev += 1
vec = x[0] / x[1] * exp(-(self.b - x[2]) ** 2 / (2 * x[1] ** 2))
return sum((self.a - vec) ** 2)
class EggCrate(Benchmark):
r"""
Egg Crate objective function.
This class defines the Egg Crate [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{EggCrate}}(x) = x_1^2 + x_2^2 + 25 \left[ \sin^2(x_1)
+ \sin^2(x_2) \right]
with :math:`x_i \in [-5, 5]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
self.global_optimum = [[0.0, 0.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return x[0] ** 2 + x[1] ** 2 + 25 * (sin(x[0]) ** 2 + sin(x[1]) ** 2)
class EggHolder(Benchmark):
r"""
Egg Holder [1]_ objective function.
This class defines the Egg Holder global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{EggHolder}}=\sum_{1}^{n - 1}\left[-\left(x_{i + 1}
+ 47 \right ) \sin\sqrt{\lvert x_{i+1} + x_i/2 + 47 \rvert}
- x_i \sin\sqrt{\lvert x_i - (x_{i + 1} + 47)\rvert}\right ]
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-512, 512]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = -959.640662711` for
:math:`{x} = [512, 404.2319]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: Jamil is missing a minus sign on the fglob value
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-512.1] * self.N,
[512.0] * self.N))
self.global_optimum = [[512.0, 404.2319]]
self.fglob = -959.640662711
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
vec = (-(x[1:] + 47) * sin(sqrt(abs(x[1:] + x[:-1] / 2. + 47)))
- x[:-1] * sin(sqrt(abs(x[:-1] - (x[1:] + 47)))))
return sum(vec)
class ElAttarVidyasagarDutta(Benchmark):
r"""
El-Attar-Vidyasagar-Dutta [1]_ objective function.
This class defines the El-Attar-Vidyasagar-Dutta function global
optimization problem. This is a multimodal minimization problem defined as
follows:
.. math::
f_{\text{ElAttarVidyasagarDutta}}(x) = (x_1^2 + x_2 - 10)^2
+ (x_1 + x_2^2 - 7)^2 + (x_1^2 + x_2^3 - 1)^2
with :math:`x_i \in [-100, 100]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 1.712780354` for
:math:`x= [3.40918683, -2.17143304]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-100.0] * self.N,
[100.0] * self.N))
self.custom_bounds = [(-4, 4), (-4, 4)]
self.global_optimum = [[3.40918683, -2.17143304]]
self.fglob = 1.712780354
def fun(self, x, *args):
self.nfev += 1
return ((x[0] ** 2 + x[1] - 10) ** 2 + (x[0] + x[1] ** 2 - 7) ** 2
+ (x[0] ** 2 + x[1] ** 3 - 1) ** 2)
class Exp2(Benchmark):
r"""
Exp2 objective function.
This class defines the Exp2 global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Exp2}}(x) = \sum_{i=0}^9 \left ( e^{-ix_1/10} - 5e^{-ix_2/10}
- e^{-i/10} + 5e^{-i} \right )^2
with :math:`x_i \in [0, 20]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 10.]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [20.0] * self.N))
self.custom_bounds = [(0, 2), (0, 20)]
self.global_optimum = [[1.0, 10.]]
self.fglob = 0.
def fun(self, x, *args):
self.nfev += 1
i = arange(10.)
vec = (exp(-i * x[0] / 10.) - 5 * exp(-i * x[1] / 10.) - exp(-i / 10.)
+ 5 * exp(-i)) ** 2
return sum(vec)
class Exponential(Benchmark):
r"""
Exponential [1] objective function.
This class defines the Exponential global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Exponential}}(x) = -e^{-0.5 \sum_{i=1}^n x_i^2}
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-1, 1]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x_i) = -1` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO Jamil are missing a minus sign on fglob
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-1.0] * self.N, [1.0] * self.N))
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = -1.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return -exp(-0.5 * sum(x ** 2.0))

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# -*- coding: utf-8 -*-
from .go_benchmark import Benchmark
class FreudensteinRoth(Benchmark):
r"""
FreudensteinRoth objective function.
This class defines the Freudenstein & Roth [1]_ global optimization problem.
This is a multimodal minimization problem defined as follows:
.. math::
f_{\text{FreudensteinRoth}}(x) = \left\{x_1 - 13 + \left[(5 - x_2) x_2
- 2 \right] x_2 \right\}^2 + \left \{x_1 - 29
+ \left[(x_2 + 1) x_2 - 14 \right] x_2 \right\}^2
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [5, 4]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(-3, 3), (-5, 5)]
self.global_optimum = [[5.0, 4.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
f1 = (-13.0 + x[0] + ((5.0 - x[1]) * x[1] - 2.0) * x[1]) ** 2
f2 = (-29.0 + x[0] + ((x[1] + 1.0) * x[1] - 14.0) * x[1]) ** 2
return f1 + f2

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# -*- coding: utf-8 -*-
import numpy as np
from numpy import abs, sin, cos, exp, floor, log, arange, prod, sqrt, sum
from .go_benchmark import Benchmark
class Gear(Benchmark):
r"""
Gear objective function.
This class defines the Gear [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Gear}}({x}) = \left \{ \frac{1.0}{6.931}
- \frac{\lfloor x_1\rfloor \lfloor x_2 \rfloor }
{\lfloor x_3 \rfloor \lfloor x_4 \rfloor } \right\}^2
with :math:`x_i \in [12, 60]` for :math:`i = 1, ..., 4`.
*Global optimum*: :math:`f(x) = 2.7 \cdot 10^{-12}` for :math:`x =
[16, 19, 43, 49]`, where the various :math:`x_i` may be permuted.
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=4):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([12.0] * self.N, [60.0] * self.N))
self.global_optimum = [[16, 19, 43, 49]]
self.fglob = 2.7e-12
def fun(self, x, *args):
self.nfev += 1
return (1. / 6.931
- floor(x[0]) * floor(x[1]) / floor(x[2]) / floor(x[3])) ** 2
class Giunta(Benchmark):
r"""
Giunta objective function.
This class defines the Giunta [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Giunta}}({x}) = 0.6 + \sum_{i=1}^{n} \left[\sin^{2}\left(1
- \frac{16}{15} x_i\right) - \frac{1}{50} \sin\left(4
- \frac{64}{15} x_i\right) - \sin\left(1
- \frac{16}{15} x_i\right)\right]
with :math:`x_i \in [-1, 1]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0.06447042053690566` for
:math:`x = [0.4673200277395354, 0.4673200169591304]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO Jamil has the wrong fglob. I think there is a lower value.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-1.0] * self.N, [1.0] * self.N))
self.global_optimum = [[0.4673200277395354, 0.4673200169591304]]
self.fglob = 0.06447042053690566
def fun(self, x, *args):
self.nfev += 1
arg = 16 * x / 15.0 - 1
return 0.6 + sum(sin(arg) + sin(arg) ** 2 + sin(4 * arg) / 50.)
class GoldsteinPrice(Benchmark):
r"""
Goldstein-Price objective function.
This class defines the Goldstein-Price [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{GoldsteinPrice}}(x) = \left[ 1 + (x_1 + x_2 + 1)^2
(19 - 14 x_1 + 3 x_1^2 - 14 x_2 + 6 x_1 x_2 + 3 x_2^2) \right]
\left[ 30 + ( 2x_1 - 3 x_2)^2 (18 - 32 x_1 + 12 x_1^2
+ 48 x_2 - 36 x_1 x_2 + 27 x_2^2) \right]
with :math:`x_i \in [-2, 2]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 3` for :math:`x = [0, -1]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-2.0] * self.N, [2.0] * self.N))
self.global_optimum = [[0., -1.]]
self.fglob = 3.0
def fun(self, x, *args):
self.nfev += 1
a = (1 + (x[0] + x[1] + 1) ** 2
* (19 - 14 * x[0] + 3 * x[0] ** 2
- 14 * x[1] + 6 * x[0] * x[1] + 3 * x[1] ** 2))
b = (30 + (2 * x[0] - 3 * x[1]) ** 2
* (18 - 32 * x[0] + 12 * x[0] ** 2
+ 48 * x[1] - 36 * x[0] * x[1] + 27 * x[1] ** 2))
return a * b
class Griewank(Benchmark):
r"""
Griewank objective function.
This class defines the Griewank global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Griewank}}(x) = \frac{1}{4000}\sum_{i=1}^n x_i^2
- \prod_{i=1}^n\cos\left(\frac{x_i}{\sqrt{i}}\right) + 1
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-600, 600]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-100.0] * self.N,
[100.0] * self.N))
self.custom_bounds = [(-50, 50), (-50, 50)]
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
i = arange(1., np.size(x) + 1.)
return sum(x ** 2 / 4000) - prod(cos(x / sqrt(i))) + 1
class Gulf(Benchmark):
r"""
Gulf objective function.
This class defines the Gulf [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Gulf}}(x) = \sum_{i=1}^99 \left( e^{-\frac{\lvert y_i
- x_2 \rvert^{x_3}}{x_1}} - t_i \right)
Where, in this exercise:
.. math::
t_i = i/100 \\
y_i = 25 + [-50 \log(t_i)]^{2/3}
with :math:`x_i \in [0, 60]` for :math:`i = 1, 2, 3`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [50, 25, 1.5]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO Gavana has absolute of (u - x[1]) term. Jamil doesn't... Leaving it in.
"""
def __init__(self, dimensions=3):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [50.0] * self.N))
self.global_optimum = [[50.0, 25.0, 1.5]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
m = 99.
i = arange(1., m + 1)
u = 25 + (-50 * log(i / 100.)) ** (2 / 3.)
vec = (exp(-((abs(u - x[1])) ** x[2] / x[0])) - i / 100.)
return sum(vec ** 2)

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# -*- coding: utf-8 -*-
import numpy as np
from numpy import abs, arctan2, asarray, cos, exp, arange, pi, sin, sqrt, sum
from .go_benchmark import Benchmark
class Hansen(Benchmark):
r"""
Hansen objective function.
This class defines the Hansen [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Hansen}}(x) = \left[ \sum_{i=0}^4(i+1)\cos(ix_1+i+1)\right ]
\left[\sum_{j=0}^4(j+1)\cos[(j+2)x_2+j+1])\right ]
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -176.54179` for
:math:`x = [-7.58989583, -7.70831466]`.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO Jamil #61 is missing the starting value of i.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[-7.58989583, -7.70831466]]
self.fglob = -176.54179
def fun(self, x, *args):
self.nfev += 1
i = arange(5.)
a = (i + 1) * cos(i * x[0] + i + 1)
b = (i + 1) * cos((i + 2) * x[1] + i + 1)
return sum(a) * sum(b)
class Hartmann3(Benchmark):
r"""
Hartmann3 objective function.
This class defines the Hartmann3 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Hartmann3}}(x) = -\sum\limits_{i=1}^{4} c_i
e^{-\sum\limits_{j=1}^{n}a_{ij}(x_j - p_{ij})^2}
Where, in this exercise:
.. math::
\begin{array}{l|ccc|c|ccr}
\hline
i & & a_{ij}& & c_i & & p_{ij} & \\
\hline
1 & 3.0 & 10.0 & 30.0 & 1.0 & 0.3689 & 0.1170 & 0.2673 \\
2 & 0.1 & 10.0 & 35.0 & 1.2 & 0.4699 & 0.4387 & 0.7470 \\
3 & 3.0 & 10.0 & 30.0 & 3.0 & 0.1091 & 0.8732 & 0.5547 \\
4 & 0.1 & 10.0 & 35.0 & 3.2 & 0.03815 & 0.5743 & 0.8828 \\
\hline
\end{array}
with :math:`x_i \in [0, 1]` for :math:`i = 1, 2, 3`.
*Global optimum*: :math:`f(x) = -3.8627821478`
for :math:`x = [0.11461292, 0.55564907, 0.85254697]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO Jamil #62 has an incorrect coefficient. p[1, 1] should be 0.4387
"""
def __init__(self, dimensions=3):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [1.0] * self.N))
self.global_optimum = [[0.11461292, 0.55564907, 0.85254697]]
self.fglob = -3.8627821478
self.a = asarray([[3.0, 10., 30.],
[0.1, 10., 35.],
[3.0, 10., 30.],
[0.1, 10., 35.]])
self.p = asarray([[0.3689, 0.1170, 0.2673],
[0.4699, 0.4387, 0.7470],
[0.1091, 0.8732, 0.5547],
[0.03815, 0.5743, 0.8828]])
self.c = asarray([1., 1.2, 3., 3.2])
def fun(self, x, *args):
self.nfev += 1
XX = np.atleast_2d(x)
d = sum(self.a * (XX - self.p) ** 2, axis=1)
return -sum(self.c * exp(-d))
class Hartmann6(Benchmark):
r"""
Hartmann6 objective function.
This class defines the Hartmann6 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Hartmann6}}(x) = -\sum\limits_{i=1}^{4} c_i
e^{-\sum\limits_{j=1}^{n}a_{ij}(x_j - p_{ij})^2}
Where, in this exercise:
.. math::
\begin{array}{l|cccccc|r}
\hline
i & & & a_{ij} & & & & c_i \\
\hline
1 & 10.0 & 3.0 & 17.0 & 3.50 & 1.70 & 8.00 & 1.0 \\
2 & 0.05 & 10.0 & 17.0 & 0.10 & 8.00 & 14.00 & 1.2 \\
3 & 3.00 & 3.50 & 1.70 & 10.0 & 17.00 & 8.00 & 3.0 \\
4 & 17.00 & 8.00 & 0.05 & 10.00 & 0.10 & 14.00 & 3.2 \\
\hline
\end{array}
\newline
\
\newline
\begin{array}{l|cccccr}
\hline
i & & & p_{ij} & & & \\
\hline
1 & 0.1312 & 0.1696 & 0.5569 & 0.0124 & 0.8283 & 0.5886 \\
2 & 0.2329 & 0.4135 & 0.8307 & 0.3736 & 0.1004 & 0.9991 \\
3 & 0.2348 & 0.1451 & 0.3522 & 0.2883 & 0.3047 & 0.6650 \\
4 & 0.4047 & 0.8828 & 0.8732 & 0.5743 & 0.1091 & 0.0381 \\
\hline
\end{array}
with :math:`x_i \in [0, 1]` for :math:`i = 1, ..., 6`.
*Global optimum*: :math:`f(x_i) = -3.32236801141551` for
:math:`{x} = [0.20168952, 0.15001069, 0.47687398, 0.27533243, 0.31165162,
0.65730054]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=6):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [1.0] * self.N))
self.global_optimum = [[0.20168952, 0.15001069, 0.47687398, 0.27533243,
0.31165162, 0.65730054]]
self.fglob = -3.32236801141551
self.a = asarray([[10., 3., 17., 3.5, 1.7, 8.],
[0.05, 10., 17., 0.1, 8., 14.],
[3., 3.5, 1.7, 10., 17., 8.],
[17., 8., 0.05, 10., 0.1, 14.]])
self.p = asarray([[0.1312, 0.1696, 0.5569, 0.0124, 0.8283, 0.5886],
[0.2329, 0.4135, 0.8307, 0.3736, 0.1004, 0.9991],
[0.2348, 0.1451, 0.3522, 0.2883, 0.3047, 0.665],
[0.4047, 0.8828, 0.8732, 0.5743, 0.1091, 0.0381]])
self.c = asarray([1.0, 1.2, 3.0, 3.2])
def fun(self, x, *args):
self.nfev += 1
XX = np.atleast_2d(x)
d = sum(self.a * (XX - self.p) ** 2, axis=1)
return -sum(self.c * exp(-d))
class HelicalValley(Benchmark):
r"""
HelicalValley objective function.
This class defines the HelicalValley [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{HelicalValley}}({x}) = 100{[z-10\Psi(x_1,x_2)]^2
+(\sqrt{x_1^2+x_2^2}-1)^2}+x_3^2
Where, in this exercise:
.. math::
2\pi\Psi(x,y) = \begin{cases} \arctan(y/x) & \textrm{for} x > 0 \\
\pi + \arctan(y/x) & \textrm{for } x < 0 \end{cases}
with :math:`x_i \in [-100, 100]` for :math:`i = 1, 2, 3`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 0, 0]`
.. [1] Fletcher, R. & Powell, M. A Rapidly Convergent Descent Method for
Minimzation, Computer Journal, 1963, 62, 163-168
TODO: Jamil equation is different to original reference. The above paper
can be obtained from
http://galton.uchicago.edu/~lekheng/courses/302/classics/
fletcher-powell.pdf
"""
def __init__(self, dimensions=3):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.] * self.N, [10.] * self.N))
self.global_optimum = [[1.0, 0.0, 0.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
r = sqrt(x[0] ** 2 + x[1] ** 2)
theta = 1 / (2. * pi) * arctan2(x[1], x[0])
return x[2] ** 2 + 100 * ((x[2] - 10 * theta) ** 2 + (r - 1) ** 2)
class HimmelBlau(Benchmark):
r"""
HimmelBlau objective function.
This class defines the HimmelBlau [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{HimmelBlau}}({x}) = (x_1^2 + x_2 - 11)^2 + (x_1 + x_2^2 - 7)^2
with :math:`x_i \in [-6, 6]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [3, 2]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.] * self.N, [5.] * self.N))
self.global_optimum = [[3.0, 2.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return (x[0] ** 2 + x[1] - 11) ** 2 + (x[0] + x[1] ** 2 - 7) ** 2
class HolderTable(Benchmark):
r"""
HolderTable objective function.
This class defines the HolderTable [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{HolderTable}}({x}) = - \left|{e^{\left|{1
- \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{\pi} }\right|}
\sin\left(x_{1}\right) \cos\left(x_{2}\right)}\right|
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -19.20850256788675` for
:math:`x_i = \pm 9.664590028909654` for :math:`i = 1, 2`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO: Jamil #146 equation is wrong - should be squaring the x1 and x2
terms, but isn't. Gavana does.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [(8.055023472141116, 9.664590028909654),
(-8.055023472141116, 9.664590028909654),
(8.055023472141116, -9.664590028909654),
(-8.055023472141116, -9.664590028909654)]
self.fglob = -19.20850256788675
def fun(self, x, *args):
self.nfev += 1
return -abs(sin(x[0]) * cos(x[1])
* exp(abs(1 - sqrt(x[0] ** 2 + x[1] ** 2) / pi)))
class Hosaki(Benchmark):
r"""
Hosaki objective function.
This class defines the Hosaki [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Hosaki}}(x) = \left ( 1 - 8 x_1 + 7 x_1^2 - \frac{7}{3} x_1^3
+ \frac{1}{4} x_1^4 \right ) x_2^2 e^{-x_1}
with :math:`x_i \in [0, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -2.3458115` for :math:`x = [4, 2]`.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = ([0., 5.], [0., 6.])
self.custom_bounds = [(0, 5), (0, 5)]
self.global_optimum = [[4, 2]]
self.fglob = -2.3458115
def fun(self, x, *args):
self.nfev += 1
val = (1 - 8 * x[0] + 7 * x[0] ** 2 - 7 / 3. * x[0] ** 3
+ 0.25 * x[0] ** 4)
return val * x[1] ** 2 * exp(-x[1])

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# -*- coding: utf-8 -*-
from numpy import sin, sum
from .go_benchmark import Benchmark
class Infinity(Benchmark):
r"""
Infinity objective function.
This class defines the Infinity [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Infinity}}(x) = \sum_{i=1}^{n} x_i^{6}
\left [ \sin\left ( \frac{1}{x_i} \right ) + 2 \right ]
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-1, 1]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-1.0] * self.N, [1.0] * self.N))
self.global_optimum = [[1e-16 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return sum(x ** 6.0 * (sin(1.0 / x) + 2.0))

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# -*- coding: utf-8 -*-
from numpy import sum, asarray, arange, exp
from .go_benchmark import Benchmark
class JennrichSampson(Benchmark):
r"""
Jennrich-Sampson objective function.
This class defines the Jennrich-Sampson [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{JennrichSampson}}(x) = \sum_{i=1}^{10} \left [2 + 2i
- (e^{ix_1} + e^{ix_2}) \right ]^2
with :math:`x_i \in [-1, 1]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 124.3621824` for
:math:`x = [0.257825, 0.257825]`.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-1.0] * self.N, [1.0] * self.N))
self.global_optimum = [[0.257825, 0.257825]]
self.custom_bounds = [(-1, 0.34), (-1, 0.34)]
self.fglob = 124.3621824
def fun(self, x, *args):
self.nfev += 1
i = arange(1, 11)
return sum((2 + 2 * i - (exp(i * x[0]) + exp(i * x[1]))) ** 2)
class Judge(Benchmark):
r"""
Judge objective function.
This class defines the Judge [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Judge}}(x) = \sum_{i=1}^{20}
\left [ \left (x_1 + A_i x_2 + B x_2^2 \right ) - C_i \right ]^2
Where, in this exercise:
.. math::
\begin{cases}
C = [4.284, 4.149, 3.877, 0.533, 2.211, 2.389, 2.145,
3.231, 1.998, 1.379, 2.106, 1.428, 1.011, 2.179, 2.858, 1.388, 1.651,
1.593, 1.046, 2.152] \\
A = [0.286, 0.973, 0.384, 0.276, 0.973, 0.543, 0.957, 0.948, 0.543,
0.797, 0.936, 0.889, 0.006, 0.828, 0.399, 0.617, 0.939, 0.784,
0.072, 0.889] \\
B = [0.645, 0.585, 0.310, 0.058, 0.455, 0.779, 0.259, 0.202, 0.028,
0.099, 0.142, 0.296, 0.175, 0.180, 0.842, 0.039, 0.103, 0.620,
0.158, 0.704]
\end{cases}
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x_i) = 16.0817307` for
:math:`\mathbf{x} = [0.86479, 1.2357]`.
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[0.86479, 1.2357]]
self.custom_bounds = [(-2.0, 2.0), (-2.0, 2.0)]
self.fglob = 16.0817307
self.c = asarray([4.284, 4.149, 3.877, 0.533, 2.211, 2.389, 2.145,
3.231, 1.998, 1.379, 2.106, 1.428, 1.011, 2.179,
2.858, 1.388, 1.651, 1.593, 1.046, 2.152])
self.a = asarray([0.286, 0.973, 0.384, 0.276, 0.973, 0.543, 0.957,
0.948, 0.543, 0.797, 0.936, 0.889, 0.006, 0.828,
0.399, 0.617, 0.939, 0.784, 0.072, 0.889])
self.b = asarray([0.645, 0.585, 0.310, 0.058, 0.455, 0.779, 0.259,
0.202, 0.028, 0.099, 0.142, 0.296, 0.175, 0.180,
0.842, 0.039, 0.103, 0.620, 0.158, 0.704])
def fun(self, x, *args):
self.nfev += 1
return sum(((x[0] + x[1] * self.a + (x[1] ** 2.0) * self.b) - self.c)
** 2.0)

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# -*- coding: utf-8 -*-
from numpy import asarray, atleast_2d, arange, sin, sqrt, prod, sum, round
from .go_benchmark import Benchmark
class Katsuura(Benchmark):
r"""
Katsuura objective function.
This class defines the Katsuura [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Katsuura}}(x) = \prod_{i=0}^{n-1} \left [ 1 +
(i+1) \sum_{k=1}^{d} \lfloor (2^k x_i) \rfloor 2^{-k} \right ]
Where, in this exercise, :math:`d = 32`.
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [0, 100]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 1` for :math:`x_i = 0` for
:math:`i = 1, ..., n`.
.. [1] Adorio, E. MVF - "Multivariate Test Functions Library in C for
Unconstrained Global Optimization", 2005
.. [2] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO: Adorio has wrong global minimum. Adorio uses round, Gavana docstring
uses floor, but Gavana code uses round. We'll use round...
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [100.0] * self.N))
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.custom_bounds = [(0, 1), (0, 1)]
self.fglob = 1.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
d = 32
k = atleast_2d(arange(1, d + 1)).T
i = arange(0., self.N * 1.)
inner = round(2 ** k * x) * (2. ** (-k))
return prod(sum(inner, axis=0) * (i + 1) + 1)
class Keane(Benchmark):
r"""
Keane objective function.
This class defines the Keane [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Keane}}(x) = \frac{\sin^2(x_1 - x_2)\sin^2(x_1 + x_2)}
{\sqrt{x_1^2 + x_2^2}}
with :math:`x_i \in [0, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0.0` for
:math:`x = [7.85396153, 7.85396135]`.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: Jamil #69, there is no way that the function can have a negative
value. Everything is squared. I think that they have the wrong solution.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [10.0] * self.N))
self.global_optimum = [[7.85396153, 7.85396135]]
self.custom_bounds = [(-1, 0.34), (-1, 0.34)]
self.fglob = 0.
def fun(self, x, *args):
self.nfev += 1
val = sin(x[0] - x[1]) ** 2 * sin(x[0] + x[1]) ** 2
return val / sqrt(x[0] ** 2 + x[1] ** 2)
class Kowalik(Benchmark):
r"""
Kowalik objective function.
This class defines the Kowalik [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Kowalik}}(x) = \sum_{i=0}^{10} \left [ a_i
- \frac{x_1 (b_i^2 + b_i x_2)} {b_i^2 + b_i x_3 + x_4} \right ]^2
Where:
.. math::
\begin{matrix}
a = [4, 2, 1, 1/2, 1/4 1/8, 1/10, 1/12, 1/14, 1/16] \\
b = [0.1957, 0.1947, 0.1735, 0.1600, 0.0844, 0.0627,
0.0456, 0.0342, 0.0323, 0.0235, 0.0246]\\
\end{matrix}
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-5, 5]` for :math:`i = 1, ..., 4`.
*Global optimum*: :math:`f(x) = 0.00030748610` for :math:`x =
[0.192833, 0.190836, 0.123117, 0.135766]`.
..[1] https://www.itl.nist.gov/div898/strd/nls/data/mgh09.shtml
"""
def __init__(self, dimensions=4):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
self.global_optimum = [[0.192833, 0.190836, 0.123117, 0.135766]]
self.fglob = 0.00030748610
self.a = asarray([4.0, 2.0, 1.0, 1 / 2.0, 1 / 4.0, 1 / 6.0, 1 / 8.0,
1 / 10.0, 1 / 12.0, 1 / 14.0, 1 / 16.0])
self.b = asarray([0.1957, 0.1947, 0.1735, 0.1600, 0.0844, 0.0627,
0.0456, 0.0342, 0.0323, 0.0235, 0.0246])
def fun(self, x, *args):
self.nfev += 1
vec = self.b - (x[0] * (self.a ** 2 + self.a * x[1])
/ (self.a ** 2 + self.a * x[2] + x[3]))
return sum(vec ** 2)

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# -*- coding: utf-8 -*-
from numpy import sum, cos, exp, pi, arange, sin
from .go_benchmark import Benchmark
class Langermann(Benchmark):
r"""
Langermann objective function.
This class defines the Langermann [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Langermann}}(x) = - \sum_{i=1}^{5}
\frac{c_i \cos\left\{\pi \left[\left(x_{1}- a_i\right)^{2}
+ \left(x_{2} - b_i \right)^{2}\right]\right\}}{e^{\frac{\left( x_{1}
- a_i\right)^{2} + \left( x_{2} - b_i\right)^{2}}{\pi}}}
Where:
.. math::
\begin{matrix}
a = [3, 5, 2, 1, 7]\\
b = [5, 2, 1, 4, 9]\\
c = [1, 2, 5, 2, 3] \\
\end{matrix}
Here :math:`x_i \in [0, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -5.1621259`
for :math:`x = [2.00299219, 1.006096]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO: Langermann from Gavana is _not the same_ as Jamil #68.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [10.0] * self.N))
self.global_optimum = [[2.00299219, 1.006096]]
self.fglob = -5.1621259
def fun(self, x, *args):
self.nfev += 1
a = [3, 5, 2, 1, 7]
b = [5, 2, 1, 4, 9]
c = [1, 2, 5, 2, 3]
return (-sum(c * exp(-(1 / pi) * ((x[0] - a) ** 2 +
(x[1] - b) ** 2)) * cos(pi * ((x[0] - a) ** 2
+ (x[1] - b) ** 2))))
class LennardJones(Benchmark):
r"""
LennardJones objective function.
This class defines the Lennard-Jones global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{LennardJones}}(\mathbf{x}) = \sum_{i=0}^{n-2}\sum_{j>1}^{n-1}
\frac{1}{r_{ij}^{12}} - \frac{1}{r_{ij}^{6}}
Where, in this exercise:
.. math::
r_{ij} = \sqrt{(x_{3i}-x_{3j})^2 + (x_{3i+1}-x_{3j+1})^2)
+ (x_{3i+2}-x_{3j+2})^2}
Valid for any dimension, :math:`n = 3*k, k=2 , 3, 4, ..., 20`. :math:`k`
is the number of atoms in 3-D space constraints: unconstrained type:
multi-modal with one global minimum; non-separable
Value-to-reach: :math:`minima[k-2] + 0.0001`. See array of minima below;
additional minima available at the Cambridge cluster database:
http://www-wales.ch.cam.ac.uk/~jon/structures/LJ/tables.150.html
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-4, 4]` for :math:`i = 1 ,..., n`.
*Global optimum*:
.. math::
\text{minima} = [-1.,-3.,-6.,-9.103852,-12.712062,-16.505384,\\
-19.821489, -24.113360, -28.422532,-32.765970,\\
-37.967600,-44.326801, -47.845157,-52.322627,\\
-56.815742,-61.317995, -66.530949, -72.659782,\\
-77.1777043]\\
"""
def __init__(self, dimensions=6):
# dimensions is in [6:60]
# max dimensions is going to be 60.
if dimensions not in range(6, 61):
raise ValueError("LJ dimensions must be in (6, 60)")
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-4.0] * self.N, [4.0] * self.N))
self.global_optimum = [[]]
self.minima = [-1.0, -3.0, -6.0, -9.103852, -12.712062,
-16.505384, -19.821489, -24.113360, -28.422532,
-32.765970, -37.967600, -44.326801, -47.845157,
-52.322627, -56.815742, -61.317995, -66.530949,
-72.659782, -77.1777043]
k = int(dimensions / 3)
self.fglob = self.minima[k - 2]
self.change_dimensionality = True
def change_dimensions(self, ndim):
if ndim not in range(6, 61):
raise ValueError("LJ dimensions must be in (6, 60)")
Benchmark.change_dimensions(self, ndim)
self.fglob = self.minima[int(self.N / 3) - 2]
def fun(self, x, *args):
self.nfev += 1
k = int(self.N / 3)
s = 0.0
for i in range(k - 1):
for j in range(i + 1, k):
a = 3 * i
b = 3 * j
xd = x[a] - x[b]
yd = x[a + 1] - x[b + 1]
zd = x[a + 2] - x[b + 2]
ed = xd * xd + yd * yd + zd * zd
ud = ed * ed * ed
if ed > 0.0:
s += (1.0 / ud - 2.0) / ud
return s
class Leon(Benchmark):
r"""
Leon objective function.
This class defines the Leon [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Leon}}(\mathbf{x}) = \left(1 - x_{1}\right)^{2}
+ 100 \left(x_{2} - x_{1}^{2} \right)^{2}
with :math:`x_i \in [-1.2, 1.2]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 1]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-1.2] * self.N, [1.2] * self.N))
self.global_optimum = [[1 for _ in range(self.N)]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return 100. * (x[1] - x[0] ** 2.0) ** 2.0 + (1 - x[0]) ** 2.0
class Levy03(Benchmark):
r"""
Levy 3 objective function.
This class defines the Levy 3 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Levy03}}(\mathbf{x}) = \sin^2(\pi y_1)+\sum_{i=1}^{n-1}(y_i-1)^2[1+10\sin^2(\pi y_{i+1})]+(y_n-1)^2
Where, in this exercise:
.. math::
y_i=1+\frac{x_i-1}{4}
Here, :math:`n` represents the number of dimensions and :math:`x_i \in [-10, 10]` for :math:`i=1,...,n`.
*Global optimum*: :math:`f(x_i) = 0` for :math:`x_i = 1` for :math:`i=1,...,n`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO: not clear what the Levy function definition is. Gavana, Mishra,
Adorio have different forms. Indeed Levy 3 docstring from Gavana
disagrees with the Gavana code! The following code is from the Mishra
listing of Levy08.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(-5, 5), (-5, 5)]
self.global_optimum = [[1 for _ in range(self.N)]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
y = 1 + (x - 1) / 4
v = sum((y[:-1] - 1) ** 2 * (1 + 10 * sin(pi * y[1:]) ** 2))
z = (y[-1] - 1) ** 2
return sin(pi * y[0]) ** 2 + v + z
class Levy05(Benchmark):
r"""
Levy 5 objective function.
This class defines the Levy 5 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Levy05}}(\mathbf{x}) = \sum_{i=1}^{5} i \cos \left[(i-1)x_1 + i \right] \times \sum_{j=1}^{5} j \cos \left[(j+1)x_2 + j \right] + (x_1 + 1.42513)^2 + (x_2 + 0.80032)^2
Here, :math:`n` represents the number of dimensions and :math:`x_i \in [-10, 10]` for :math:`i=1,...,n`.
*Global optimum*: :math:`f(x_i) = -176.1375779` for :math:`\mathbf{x} = [-1.30685, -1.42485]`.
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = ([-2.0, 2.0], [-2.0, 2.0])
self.global_optimum = [[-1.30685, -1.42485]]
self.fglob = -176.1375779
def fun(self, x, *args):
self.nfev += 1
i = arange(1, 6)
a = i * cos((i - 1) * x[0] + i)
b = i * cos((i + 1) * x[1] + i)
return sum(a) * sum(b) + (x[0] + 1.42513) ** 2 + (x[1] + 0.80032) ** 2
class Levy13(Benchmark):
r"""
Levy13 objective function.
This class defines the Levy13 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Levy13}}(x) = \left(x_{1} -1\right)^{2} \left[\sin^{2}
\left(3 \pi x_{2}\right) + 1\right] + \left(x_{2}
- 1\right)^{2} \left[\sin^{2}\left(2 \pi x_{2}\right)
+ 1\right] + \sin^{2}\left(3 \pi x_{1}\right)
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 1]`
.. [1] Mishra, S. Some new test functions for global optimization and
performance of repulsive particle swarm method.
Munich Personal RePEc Archive, 2006, 2718
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(-5, 5), (-5, 5)]
self.global_optimum = [[1 for _ in range(self.N)]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
u = sin(3 * pi * x[0]) ** 2
v = (x[0] - 1) ** 2 * (1 + (sin(3 * pi * x[1])) ** 2)
w = (x[1] - 1) ** 2 * (1 + (sin(2 * pi * x[1])) ** 2)
return u + v + w

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# -*- coding: utf-8 -*-
from numpy import (abs, asarray, cos, exp, log, arange, pi, prod, sin, sqrt,
sum, tan)
from .go_benchmark import Benchmark, safe_import
with safe_import():
from scipy.special import factorial
class Matyas(Benchmark):
r"""
Matyas objective function.
This class defines the Matyas [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Matyas}}(x) = 0.26(x_1^2 + x_2^2) - 0.48 x_1 x_2
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return 0.26 * (x[0] ** 2 + x[1] ** 2) - 0.48 * x[0] * x[1]
class McCormick(Benchmark):
r"""
McCormick objective function.
This class defines the McCormick [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{McCormick}}(x) = - x_{1} + 2 x_{2} + \left(x_{1}
- x_{2}\right)^{2} + \sin\left(x_{1} + x_{2}\right) + 1
with :math:`x_1 \in [-1.5, 4]`, :math:`x_2 \in [-3, 4]`.
*Global optimum*: :math:`f(x) = -1.913222954981037` for
:math:`x = [-0.5471975602214493, -1.547197559268372]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = [(-1.5, 4.0), (-3.0, 3.0)]
self.global_optimum = [[-0.5471975602214493, -1.547197559268372]]
self.fglob = -1.913222954981037
def fun(self, x, *args):
self.nfev += 1
return (sin(x[0] + x[1]) + (x[0] - x[1]) ** 2 - 1.5 * x[0]
+ 2.5 * x[1] + 1)
class Meyer(Benchmark):
r"""
Meyer [1]_ objective function.
..[1] https://www.itl.nist.gov/div898/strd/nls/data/mgh10.shtml
TODO NIST regression standard
"""
def __init__(self, dimensions=3):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0., 100., 100.],
[1, 1000., 500.]))
self.global_optimum = [[5.6096364710e-3, 6.1813463463e3,
3.4522363462e2]]
self.fglob = 8.7945855171e1
self.a = asarray([3.478E+04, 2.861E+04, 2.365E+04, 1.963E+04, 1.637E+04,
1.372E+04, 1.154E+04, 9.744E+03, 8.261E+03, 7.030E+03,
6.005E+03, 5.147E+03, 4.427E+03, 3.820E+03, 3.307E+03,
2.872E+03])
self.b = asarray([5.000E+01, 5.500E+01, 6.000E+01, 6.500E+01, 7.000E+01,
7.500E+01, 8.000E+01, 8.500E+01, 9.000E+01, 9.500E+01,
1.000E+02, 1.050E+02, 1.100E+02, 1.150E+02, 1.200E+02,
1.250E+02])
def fun(self, x, *args):
self.nfev += 1
vec = x[0] * exp(x[1] / (self.b + x[2]))
return sum((self.a - vec) ** 2)
class Michalewicz(Benchmark):
r"""
Michalewicz objective function.
This class defines the Michalewicz [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Michalewicz}}(x) = - \sum_{i=1}^{2} \sin\left(x_i\right)
\sin^{2 m}\left(\frac{i x_i^{2}}{\pi}\right)
Where, in this exercise, :math:`m = 10`.
with :math:`x_i \in [0, \pi]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x_i) = -1.8013` for :math:`x = [0, 0]`
.. [1] Adorio, E. MVF - "Multivariate Test Functions Library in C for
Unconstrained Global Optimization", 2005
TODO: could change dimensionality, but global minimum might change.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [pi] * self.N))
self.global_optimum = [[2.20290555, 1.570796]]
self.fglob = -1.8013
def fun(self, x, *args):
self.nfev += 1
m = 10.0
i = arange(1, self.N + 1)
return -sum(sin(x) * sin(i * x ** 2 / pi) ** (2 * m))
class MieleCantrell(Benchmark):
r"""
Miele-Cantrell [1]_ objective function.
This class defines the Miele-Cantrell global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{MieleCantrell}}({x}) = (e^{-x_1} - x_2)^4 + 100(x_2 - x_3)^6
+ \tan^4(x_3 - x_4) + x_1^8
with :math:`x_i \in [-1, 1]` for :math:`i = 1, ..., 4`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, 1, 1, 1]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=4):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-1.0] * self.N, [1.0] * self.N))
self.global_optimum = [[0.0, 1.0, 1.0, 1.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return ((exp(-x[0]) - x[1]) ** 4 + 100 * (x[1] - x[2]) ** 6
+ tan(x[2] - x[3]) ** 4 + x[0] ** 8)
class Mishra01(Benchmark):
r"""
Mishra 1 objective function.
This class defines the Mishra 1 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Mishra01}}(x) = (1 + x_n)^{x_n}
where
.. math::
x_n = n - \sum_{i=1}^{n-1} x_i
with :math:`x_i \in [0, 1]` for :math:`i =1, ..., n`.
*Global optimum*: :math:`f(x) = 2` for :math:`x_i = 1` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N,
[1.0 + 1e-9] * self.N))
self.global_optimum = [[1.0 for _ in range(self.N)]]
self.fglob = 2.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
xn = self.N - sum(x[0:-1])
return (1 + xn) ** xn
class Mishra02(Benchmark):
r"""
Mishra 2 objective function.
This class defines the Mishra 2 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Mishra02}}({x}) = (1 + x_n)^{x_n}
with
.. math::
x_n = n - \sum_{i=1}^{n-1} \frac{(x_i + x_{i+1})}{2}
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [0, 1]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 2` for :math:`x_i = 1`
for :math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N,
[1.0 + 1e-9] * self.N))
self.global_optimum = [[1.0 for _ in range(self.N)]]
self.fglob = 2.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
xn = self.N - sum((x[:-1] + x[1:]) / 2.0)
return (1 + xn) ** xn
class Mishra03(Benchmark):
r"""
Mishra 3 objective function.
This class defines the Mishra 3 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Mishra03}}(x) = \sqrt{\lvert \cos{\sqrt{\lvert x_1^2
+ x_2^2 \rvert}} \rvert} + 0.01(x_1 + x_2)
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -0.1999` for
:math:`x = [-9.99378322, -9.99918927]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: I think that Jamil#76 has the wrong global minimum, a smaller one
is possible
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[-9.99378322, -9.99918927]]
self.fglob = -0.19990562
def fun(self, x, *args):
self.nfev += 1
return ((0.01 * (x[0] + x[1])
+ sqrt(abs(cos(sqrt(abs(x[0] ** 2 + x[1] ** 2)))))))
class Mishra04(Benchmark):
r"""
Mishra 4 objective function.
This class defines the Mishra 4 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Mishra04}}({x}) = \sqrt{\lvert \sin{\sqrt{\lvert
x_1^2 + x_2^2 \rvert}} \rvert} + 0.01(x_1 + x_2)
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -0.17767` for
:math:`x = [-8.71499636, -9.0533148]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: I think that Jamil#77 has the wrong minimum, not possible
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[-8.88055269734, -8.89097599857]]
self.fglob = -0.177715264826
def fun(self, x, *args):
self.nfev += 1
return ((0.01 * (x[0] + x[1])
+ sqrt(abs(sin(sqrt(abs(x[0] ** 2 + x[1] ** 2)))))))
class Mishra05(Benchmark):
r"""
Mishra 5 objective function.
This class defines the Mishra 5 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Mishra05}}(x) = \left [ \sin^2 ((\cos(x_1) + \cos(x_2))^2)
+ \cos^2 ((\sin(x_1) + \sin(x_2))^2) + x_1 \right ]^2 + 0.01(x_1 + x_2)
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -0.119829` for :math:`x = [-1.98682, -10]`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO Line 381 in paper
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[-1.98682, -10.0]]
self.fglob = -1.019829519930646
def fun(self, x, *args):
self.nfev += 1
return (0.01 * x[0] + 0.1 * x[1]
+ (sin((cos(x[0]) + cos(x[1])) ** 2) ** 2
+ cos((sin(x[0]) + sin(x[1])) ** 2) ** 2 + x[0]) ** 2)
class Mishra06(Benchmark):
r"""
Mishra 6 objective function.
This class defines the Mishra 6 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Mishra06}}(x) = -\log{\left [ \sin^2 ((\cos(x_1)
+ \cos(x_2))^2) - \cos^2 ((\sin(x_1) + \sin(x_2))^2) + x_1 \right ]^2}
+ 0.01 \left[(x_1 -1)^2 + (x_2 - 1)^2 \right]
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x_i) = -2.28395` for :math:`x = [2.88631, 1.82326]`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO line 397
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[2.88631, 1.82326]]
self.fglob = -2.28395
def fun(self, x, *args):
self.nfev += 1
a = 0.1 * ((x[0] - 1) ** 2 + (x[1] - 1) ** 2)
u = (cos(x[0]) + cos(x[1])) ** 2
v = (sin(x[0]) + sin(x[1])) ** 2
return a - log((sin(u) ** 2 - cos(v) ** 2 + x[0]) ** 2)
class Mishra07(Benchmark):
r"""
Mishra 7 objective function.
This class defines the Mishra 7 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Mishra07}}(x) = \left [\prod_{i=1}^{n} x_i - n! \right]^2
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = \sqrt{n}`
for :math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(-2, 2), (-2, 2)]
self.global_optimum = [[sqrt(self.N)
for i in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return (prod(x) - factorial(self.N)) ** 2.0
class Mishra08(Benchmark):
r"""
Mishra 8 objective function.
This class defines the Mishra 8 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Mishra08}}(x) = 0.001 \left[\lvert x_1^{10} - 20x_1^9
+ 180x_1^8 - 960 x_1^7 + 3360x_1^6 - 8064x_1^5 + 13340x_1^4 - 15360x_1^3
+ 11520x_1^2 - 5120x_1 + 2624 \rvert \lvert x_2^4 + 12x_2^3 + 54x_2^2
+ 108x_2 + 81 \rvert \right]^2
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [2, -3]`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO Line 1065
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(1.0, 2.0), (-4.0, 1.0)]
self.global_optimum = [[2.0, -3.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
val = abs(x[0] ** 10 - 20 * x[0] ** 9 + 180 * x[0] ** 8
- 960 * x[0] ** 7 + 3360 * x[0] ** 6 - 8064 * x[0] ** 5
+ 13340 * x[0] ** 4 - 15360 * x[0] ** 3 + 11520 * x[0] ** 2
- 5120 * x[0] + 2624)
val += abs(x[1] ** 4 + 12 * x[1] ** 3 +
54 * x[1] ** 2 + 108 * x[1] + 81)
return 0.001 * val ** 2
class Mishra09(Benchmark):
r"""
Mishra 9 objective function.
This class defines the Mishra 9 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Mishra09}}({x}) = \left[ ab^2c + abc^2 + b^2
+ (x_1 + x_2 - x_3)^2 \right]^2
Where, in this exercise:
.. math::
\begin{cases} a = 2x_1^3 + 5x_1x_2 + 4x_3 - 2x_1^2x_3 - 18 \\
b = x_1 + x_2^3 + x_1x_2^2 + x_1x_3^2 - 22 \\
c = 8x_1^2 + 2x_2x_3 + 2x_2^2 + 3x_2^3 - 52 \end{cases}
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2, 3`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 2, 3]`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO Line 1103
"""
def __init__(self, dimensions=3):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[1.0, 2.0, 3.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
a = (2 * x[0] ** 3 + 5 * x[0] * x[1]
+ 4 * x[2] - 2 * x[0] ** 2 * x[2] - 18)
b = x[0] + x[1] ** 3 + x[0] * x[1] ** 2 + x[0] * x[2] ** 2 - 22.0
c = (8 * x[0] ** 2 + 2 * x[1] * x[2]
+ 2 * x[1] ** 2 + 3 * x[1] ** 3 - 52)
return (a * c * b ** 2 + a * b * c ** 2 + b ** 2
+ (x[0] + x[1] - x[2]) ** 2) ** 2
class Mishra10(Benchmark):
r"""
Mishra 10 objective function.
This class defines the Mishra 10 global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
TODO - int(x) should be used instead of floor(x)!!!!!
f_{\text{Mishra10}}({x}) = \left[ \lfloor x_1 \perp x_2 \rfloor -
\lfloor x_1 \rfloor - \lfloor x_2 \rfloor \right]^2
with :math:`x_i \in [-10, 10]` for :math:`i =1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [2, 2]`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO line 1115
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[2.0, 2.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
x1, x2 = int(x[0]), int(x[1])
f1 = x1 + x2
f2 = x1 * x2
return (f1 - f2) ** 2.0
class Mishra11(Benchmark):
r"""
Mishra 11 objective function.
This class defines the Mishra 11 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Mishra11}}(x) = \left [ \frac{1}{n} \sum_{i=1}^{n} \lvert x_i
\rvert - \left(\prod_{i=1}^{n} \lvert x_i \rvert \right )^{\frac{1}{n}}
\right]^2
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(-3, 3), (-3, 3)]
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
N = self.N
return ((1.0 / N) * sum(abs(x)) - (prod(abs(x))) ** 1.0 / N) ** 2.0
class MultiModal(Benchmark):
r"""
MultiModal objective function.
This class defines the MultiModal global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{MultiModal}}(x) = \left( \sum_{i=1}^n \lvert x_i \rvert
\right) \left( \prod_{i=1}^n \lvert x_i \rvert \right)
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(-5, 5), (-5, 5)]
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return sum(abs(x)) * prod(abs(x))

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# -*- coding: utf-8 -*-
from numpy import cos, sqrt, sin, abs
from .go_benchmark import Benchmark
class NeedleEye(Benchmark):
r"""
NeedleEye objective function.
This class defines the Needle-Eye [1]_ global optimization problem. This is a
a multimodal minimization problem defined as follows:
.. math::
f_{\text{NeedleEye}}(x) =
\begin{cases}
1 & \textrm{if }\hspace{5pt} \lvert x_i \rvert < eye \hspace{5pt}
\forall i \\
\sum_{i=1}^n (100 + \lvert x_i \rvert) & \textrm{if } \hspace{5pt}
\lvert x_i \rvert > eye \\
0 & \textrm{otherwise}\\
\end{cases}
Where, in this exercise, :math:`eye = 0.0001`.
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 1` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = 1.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
f = fp = 0.0
eye = 0.0001
for val in x:
if abs(val) >= eye:
fp = 1.0
f += 100.0 + abs(val)
else:
f += 1.0
if fp < 1e-6:
f = f / self.N
return f
class NewFunction01(Benchmark):
r"""
NewFunction01 objective function.
This class defines the NewFunction01 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{NewFunction01}}(x) = \left | {\cos\left(\sqrt{\left|{x_{1}^{2}
+ x_{2}}\right|}\right)} \right |^{0.5} + (x_{1} + x_{2})/100
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -0.18459899925` for
:math:`x = [-8.46669057, -9.99982177]`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO line 355
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[-8.46668984648, -9.99980944557]]
self.fglob = -0.184648852475
def fun(self, x, *args):
self.nfev += 1
return ((abs(cos(sqrt(abs(x[0] ** 2 + x[1]))))) ** 0.5
+ 0.01 * (x[0] + x[1]))
class NewFunction02(Benchmark):
r"""
NewFunction02 objective function.
This class defines the NewFunction02 global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{NewFunction02}}(x) = \left | {\sin\left(\sqrt{\lvert{x_{1}^{2}
+ x_{2}}\rvert}\right)} \right |^{0.5} + (x_{1} + x_{2})/100
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -0.19933159253` for
:math:`x = [-9.94103375, -9.99771235]`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO Line 368
TODO WARNING, minimum value is estimated from running many optimisations and
choosing the best.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[-9.94114736324, -9.99997128772]]
self.fglob = -0.199409030092
def fun(self, x, *args):
self.nfev += 1
return ((abs(sin(sqrt(abs(x[0] ** 2 + x[1]))))) ** 0.5
+ 0.01 * (x[0] + x[1]))
#Newfunction 3 from Gavana is entered as Mishra05.

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# -*- coding: utf-8 -*-
from numpy import sum, cos, exp, pi, asarray
from .go_benchmark import Benchmark
class OddSquare(Benchmark):
r"""
Odd Square objective function.
This class defines the Odd Square [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{OddSquare}}(x) = -e^{-\frac{d}{2\pi}} \cos(\pi d)
\left( 1 + \frac{0.02h}{d + 0.01} \right )
Where, in this exercise:
.. math::
\begin{cases}
d = n \cdot \smash{\displaystyle\max_{1 \leq i \leq n}}
\left[ (x_i - b_i)^2 \right ] \\
h = \sum_{i=1}^{n} (x_i - b_i)^2
\end{cases}
And :math:`b = [1, 1.3, 0.8, -0.4, -1.3, 1.6, -0.2, -0.6, 0.5, 1.4, 1, 1.3,
0.8, -0.4, -1.3, 1.6, -0.2, -0.6, 0.5, 1.4]`
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-5 \pi, 5 \pi]` for :math:`i = 1, ..., n` and
:math:`n \leq 20`.
*Global optimum*: :math:`f(x_i) = -1.0084` for :math:`x \approx b`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO The best solution changes on dimensionality
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0 * pi] * self.N,
[5.0 * pi] * self.N))
self.custom_bounds = ([-2.0, 4.0], [-2.0, 4.0])
self.a = asarray([1, 1.3, 0.8, -0.4, -1.3, 1.6, -0.2, -0.6, 0.5, 1.4]
* 2)
self.global_optimum = [[1.0873320463871847, 1.3873320456818079]]
self.fglob = -1.00846728102
def fun(self, x, *args):
self.nfev += 1
b = self.a[0: self.N]
d = self.N * max((x - b) ** 2.0)
h = sum((x - b) ** 2.0)
return (-exp(-d / (2.0 * pi)) * cos(pi * d)
* (1.0 + 0.02 * h / (d + 0.01)))

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# -*- coding: utf-8 -*-
from numpy import (abs, sum, sin, cos, sqrt, log, prod, where, pi, exp, arange,
floor, log10, atleast_2d, zeros)
from .go_benchmark import Benchmark
class Parsopoulos(Benchmark):
r"""
Parsopoulos objective function.
This class defines the Parsopoulos [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Parsopoulos}}(x) = \cos(x_1)^2 + \sin(x_2)^2
with :math:`x_i \in [-5, 5]` for :math:`i = 1, 2`.
*Global optimum*: This function has infinite number of global minima in R2,
at points :math:`\left(k\frac{\pi}{2}, \lambda \pi \right)`,
where :math:`k = \pm1, \pm3, ...` and :math:`\lambda = 0, \pm1, \pm2, ...`
In the given domain problem, function has 12 global minima all equal to
zero.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
self.global_optimum = [[pi / 2.0, pi]]
self.fglob = 0
def fun(self, x, *args):
self.nfev += 1
return cos(x[0]) ** 2.0 + sin(x[1]) ** 2.0
class Pathological(Benchmark):
r"""
Pathological objective function.
This class defines the Pathological [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Pathological}}(x) = \sum_{i=1}^{n -1} \frac{\sin^{2}\left(
\sqrt{100 x_{i+1}^{2} + x_{i}^{2}}\right) -0.5}{0.001 \left(x_{i}^{2}
- 2x_{i}x_{i+1} + x_{i+1}^{2}\right)^{2} + 0.50}
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-100, 100]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0.` for :math:`x = [0, 0]` for
:math:`i = 1, 2`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-100.0] * self.N,
[100.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.
def fun(self, x, *args):
self.nfev += 1
vec = (0.5 + (sin(sqrt(100 * x[: -1] ** 2 + x[1:] ** 2)) ** 2 - 0.5) /
(1. + 0.001 * (x[: -1] ** 2 - 2 * x[: -1] * x[1:]
+ x[1:] ** 2) ** 2))
return sum(vec)
class Paviani(Benchmark):
r"""
Paviani objective function.
This class defines the Paviani [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Paviani}}(x) = \sum_{i=1}^{10} \left[\log^{2}\left(10
- x_i\right) + \log^{2}\left(x_i -2\right)\right]
- \left(\prod_{i=1}^{10} x_i^{10} \right)^{0.2}
with :math:`x_i \in [2.001, 9.999]` for :math:`i = 1, ... , 10`.
*Global optimum*: :math:`f(x_i) = -45.7784684040686` for
:math:`x_i = 9.350266` for :math:`i = 1, ..., 10`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: think Gavana web/code definition is wrong because final product term
shouldn't raise x to power 10.
"""
def __init__(self, dimensions=10):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([2.001] * self.N, [9.999] * self.N))
self.global_optimum = [[9.350266 for _ in range(self.N)]]
self.fglob = -45.7784684040686
def fun(self, x, *args):
self.nfev += 1
return sum(log(x - 2) ** 2.0 + log(10.0 - x) ** 2.0) - prod(x) ** 0.2
class Penalty01(Benchmark):
r"""
Penalty 1 objective function.
This class defines the Penalty 1 [1]_ global optimization problem. This is a
imultimodal minimization problem defined as follows:
.. math::
f_{\text{Penalty01}}(x) = \frac{\pi}{30} \left\{10 \sin^2(\pi y_1)
+ \sum_{i=1}^{n-1} (y_i - 1)^2 \left[1 + 10 \sin^2(\pi y_{i+1}) \right]
+ (y_n - 1)^2 \right \} + \sum_{i=1}^n u(x_i, 10, 100, 4)
Where, in this exercise:
.. math::
y_i = 1 + \frac{1}{4}(x_i + 1)
And:
.. math::
u(x_i, a, k, m) =
\begin{cases}
k(x_i - a)^m & \textrm{if} \hspace{5pt} x_i > a \\
0 & \textrm{if} \hspace{5pt} -a \leq x_i \leq a \\
k(-x_i - a)^m & \textrm{if} \hspace{5pt} x_i < -a
\end{cases}
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-50, 50]` for :math:`i= 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = -1` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-50.0] * self.N, [50.0] * self.N))
self.custom_bounds = ([-5.0, 5.0], [-5.0, 5.0])
self.global_optimum = [[-1.0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
a, b, c = 10.0, 100.0, 4.0
xx = abs(x)
u = where(xx > a, b * (xx - a) ** c, 0.0)
y = 1.0 + (x + 1.0) / 4.0
return (sum(u) + (pi / 30.0) * (10.0 * sin(pi * y[0]) ** 2.0
+ sum((y[: -1] - 1.0) ** 2.0
* (1.0 + 10.0 * sin(pi * y[1:]) ** 2.0))
+ (y[-1] - 1) ** 2.0))
class Penalty02(Benchmark):
r"""
Penalty 2 objective function.
This class defines the Penalty 2 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Penalty02}}(x) = 0.1 \left\{\sin^2(3\pi x_1) + \sum_{i=1}^{n-1}
(x_i - 1)^2 \left[1 + \sin^2(3\pi x_{i+1}) \right ]
+ (x_n - 1)^2 \left [1 + \sin^2(2 \pi x_n) \right ]\right \}
+ \sum_{i=1}^n u(x_i, 5, 100, 4)
Where, in this exercise:
.. math::
u(x_i, a, k, m) =
\begin{cases}
k(x_i - a)^m & \textrm{if} \hspace{5pt} x_i > a \\
0 & \textrm{if} \hspace{5pt} -a \leq x_i \leq a \\
k(-x_i - a)^m & \textrm{if} \hspace{5pt} x_i < -a \\
\end{cases}
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-50, 50]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 1` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-50.0] * self.N, [50.0] * self.N))
self.custom_bounds = ([-4.0, 4.0], [-4.0, 4.0])
self.global_optimum = [[1.0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
a, b, c = 5.0, 100.0, 4.0
xx = abs(x)
u = where(xx > a, b * (xx - a) ** c, 0.0)
return (sum(u) + 0.1 * (10 * sin(3.0 * pi * x[0]) ** 2.0
+ sum((x[:-1] - 1.0) ** 2.0
* (1.0 + sin(3 * pi * x[1:]) ** 2.0))
+ (x[-1] - 1) ** 2.0 * (1 + sin(2 * pi * x[-1]) ** 2.0)))
class PenHolder(Benchmark):
r"""
PenHolder objective function.
This class defines the PenHolder [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{PenHolder}}(x) = -e^{\left|{e^{-\left|{- \frac{\sqrt{x_{1}^{2}
+ x_{2}^{2}}}{\pi} + 1}\right|} \cos\left(x_{1}\right)
\cos\left(x_{2}\right)}\right|^{-1}}
with :math:`x_i \in [-11, 11]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x_i) = -0.9635348327265058` for
:math:`x_i = \pm 9.646167671043401` for :math:`i = 1, 2`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-11.0] * self.N, [11.0] * self.N))
self.global_optimum = [[-9.646167708023526, 9.646167671043401]]
self.fglob = -0.9635348327265058
def fun(self, x, *args):
self.nfev += 1
a = abs(1. - (sqrt(x[0] ** 2 + x[1] ** 2) / pi))
b = cos(x[0]) * cos(x[1]) * exp(a)
return -exp(-abs(b) ** -1)
class PermFunction01(Benchmark):
r"""
PermFunction 1 objective function.
This class defines the PermFunction1 [1]_ global optimization problem. This is
a multimodal minimization problem defined as follows:
.. math::
f_{\text{PermFunction01}}(x) = \sum_{k=1}^n \left\{ \sum_{j=1}^n (j^k
+ \beta) \left[ \left(\frac{x_j}{j}\right)^k - 1 \right] \right\}^2
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-n, n + 1]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = i` for
:math:`i = 1, ..., n`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO: line 560
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-self.N] * self.N,
[self.N + 1] * self.N))
self.global_optimum = [list(range(1, self.N + 1))]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
b = 0.5
k = atleast_2d(arange(self.N) + 1).T
j = atleast_2d(arange(self.N) + 1)
s = (j ** k + b) * ((x / j) ** k - 1)
return sum((sum(s, axis=1) ** 2))
class PermFunction02(Benchmark):
r"""
PermFunction 2 objective function.
This class defines the Perm Function 2 [1]_ global optimization problem. This is
a multimodal minimization problem defined as follows:
.. math::
f_{\text{PermFunction02}}(x) = \sum_{k=1}^n \left\{ \sum_{j=1}^n (j
+ \beta) \left[ \left(x_j^k - {\frac{1}{j}}^{k} \right )
\right] \right\}^2
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-n, n+1]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = \frac{1}{i}`
for :math:`i = 1, ..., n`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO: line 582
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-self.N] * self.N,
[self.N + 1] * self.N))
self.custom_bounds = ([0, 1.5], [0, 1.0])
self.global_optimum = [1. / arange(1, self.N + 1)]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
b = 10
k = atleast_2d(arange(self.N) + 1).T
j = atleast_2d(arange(self.N) + 1)
s = (j + b) * (x ** k - (1. / j) ** k)
return sum((sum(s, axis=1) ** 2))
class Pinter(Benchmark):
r"""
Pinter objective function.
This class defines the Pinter [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Pinter}}(x) = \sum_{i=1}^n ix_i^2 + \sum_{i=1}^n 20i
\sin^2 A + \sum_{i=1}^n i \log_{10} (1 + iB^2)
Where, in this exercise:
.. math::
\begin{cases}
A = x_{i-1} \sin x_i + \sin x_{i+1} \\
B = x_{i-1}^2 - 2x_i + 3x_{i + 1} - \cos x_i + 1\\
\end{cases}
Where :math:`x_0 = x_n` and :math:`x_{n + 1} = x_1`.
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
i = arange(self.N) + 1
xx = zeros(self.N + 2)
xx[1: - 1] = x
xx[0] = x[-1]
xx[-1] = x[0]
A = xx[0: -2] * sin(xx[1: - 1]) + sin(xx[2:])
B = xx[0: -2] ** 2 - 2 * xx[1: - 1] + 3 * xx[2:] - cos(xx[1: - 1]) + 1
return (sum(i * x ** 2)
+ sum(20 * i * sin(A) ** 2)
+ sum(i * log10(1 + i * B ** 2)))
class Plateau(Benchmark):
r"""
Plateau objective function.
This class defines the Plateau [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Plateau}}(x) = 30 + \sum_{i=1}^n \lfloor \lvert x_i
\rvert\rfloor
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-5.12, 5.12]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 30` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.12] * self.N, [5.12] * self.N))
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = 30.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return 30.0 + sum(floor(abs(x)))
class Powell(Benchmark):
r"""
Powell objective function.
This class defines the Powell [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Powell}}(x) = (x_3+10x_1)^2 + 5(x_2-x_4)^2 + (x_1-2x_2)^4
+ 10(x_3-x_4)^4
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-4, 5]` for :math:`i = 1, ..., 4`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., 4`
..[1] Powell, M. An iterative method for finding stationary values of a
function of several variables Computer Journal, 1962, 5, 147-151
"""
def __init__(self, dimensions=4):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-4.0] * self.N, [5.0] * self.N))
self.global_optimum = [[0, 0, 0, 0]]
self.fglob = 0
def fun(self, x, *args):
self.nfev += 1
return ((x[0] + 10 * x[1]) ** 2 + 5 * (x[2] - x[3]) ** 2
+ (x[1] - 2 * x[2]) ** 4 + 10 * (x[0] - x[3]) ** 4)
class PowerSum(Benchmark):
r"""
Power sum objective function.
This class defines the Power Sum global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{PowerSum}}(x) = \sum_{k=1}^n\left[\left(\sum_{i=1}^n x_i^k
\right) - b_k \right]^2
Where, in this exercise, :math:`b = [8, 18, 44, 114]`
Here, :math:`x_i \in [0, 4]` for :math:`i = 1, ..., 4`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 2, 2, 3]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=4):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N,
[4.0] * self.N))
self.global_optimum = [[1.0, 2.0, 2.0, 3.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
b = [8.0, 18.0, 44.0, 114.0]
k = atleast_2d(arange(self.N) + 1).T
return sum((sum(x ** k, axis=1) - b) ** 2)
class Price01(Benchmark):
r"""
Price 1 objective function.
This class defines the Price 1 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Price01}}(x) = (\lvert x_1 \rvert - 5)^2
+ (\lvert x_2 \rvert - 5)^2
with :math:`x_i \in [-500, 500]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x_i) = 0.0` for :math:`x = [5, 5]` or
:math:`x = [5, -5]` or :math:`x = [-5, 5]` or :math:`x = [-5, -5]`.
.. [1] Price, W. A controlled random search procedure for global
optimisation Computer Journal, 1977, 20, 367-370
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-500.0] * self.N,
[500.0] * self.N))
self.custom_bounds = ([-10.0, 10.0], [-10.0, 10.0])
self.global_optimum = [[5.0, 5.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return (abs(x[0]) - 5.0) ** 2.0 + (abs(x[1]) - 5.0) ** 2.0
class Price02(Benchmark):
r"""
Price 2 objective function.
This class defines the Price 2 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Price02}}(x) = 1 + \sin^2(x_1) + \sin^2(x_2)
- 0.1e^{(-x_1^2 - x_2^2)}
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0.9` for :math:`x_i = [0, 0]`
.. [1] Price, W. A controlled random search procedure for global
optimisation Computer Journal, 1977, 20, 367-370
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[0.0, 0.0]]
self.fglob = 0.9
def fun(self, x, *args):
self.nfev += 1
return 1.0 + sum(sin(x) ** 2) - 0.1 * exp(-x[0] ** 2.0 - x[1] ** 2.0)
class Price03(Benchmark):
r"""
Price 3 objective function.
This class defines the Price 3 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Price03}}(x) = 100(x_2 - x_1^2)^2 + \left[6.4(x_2 - 0.5)^2
- x_1 - 0.6 \right]^2
with :math:`x_i \in [-50, 50]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [-5, -5]`,
:math:`x = [-5, 5]`, :math:`x = [5, -5]`, :math:`x = [5, 5]`.
.. [1] Price, W. A controlled random search procedure for global
optimisation Computer Journal, 1977, 20, 367-370
TODO Jamil #96 has an erroneous factor of 6 in front of the square brackets
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
self.custom_bounds = ([0, 2], [0, 2])
self.global_optimum = [[1.0, 1.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return (100 * (x[1] - x[0] ** 2) ** 2
+ (6.4 * (x[1] - 0.5) ** 2 - x[0] - 0.6) ** 2)
class Price04(Benchmark):
r"""
Price 4 objective function.
This class defines the Price 4 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Price04}}(x) = (2 x_1^3 x_2 - x_2^3)^2
+ (6 x_1 - x_2^2 + x_2)^2
with :math:`x_i \in [-50, 50]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, 0]`,
:math:`x = [2, 4]` and :math:`x = [1.464, -2.506]`
.. [1] Price, W. A controlled random search procedure for global
optimisation Computer Journal, 1977, 20, 367-370
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-50.0] * self.N, [50.0] * self.N))
self.custom_bounds = ([0, 2], [0, 2])
self.global_optimum = [[2.0, 4.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return ((2.0 * x[1] * x[0] ** 3.0 - x[1] ** 3.0) ** 2.0
+ (6.0 * x[0] - x[1] ** 2.0 + x[1]) ** 2.0)

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# -*- coding: utf-8 -*-
from numpy import abs, sum, arange, sqrt
from .go_benchmark import Benchmark
class Qing(Benchmark):
r"""
Qing objective function.
This class defines the Qing [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Qing}}(x) = \sum_{i=1}^{n} (x_i^2 - i)^2
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-500, 500]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = \pm \sqrt(i)` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-500.0] * self.N,
[500.0] * self.N))
self.custom_bounds = [(-2, 2), (-2, 2)]
self.global_optimum = [[sqrt(_) for _ in range(1, self.N + 1)]]
self.fglob = 0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
i = arange(1, self.N + 1)
return sum((x ** 2.0 - i) ** 2.0)
class Quadratic(Benchmark):
r"""
Quadratic objective function.
This class defines the Quadratic [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Quadratic}}(x) = -3803.84 - 138.08x_1 - 232.92x_2 + 128.08x_1^2
+ 203.64x_2^2 + 182.25x_1x_2
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -3873.72418` for
:math:`x = [0.19388, 0.48513]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(0, 1), (0, 1)]
self.global_optimum = [[0.19388, 0.48513]]
self.fglob = -3873.72418
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return (-3803.84 - 138.08 * x[0] - 232.92 * x[1] + 128.08 * x[0] ** 2.0
+ 203.64 * x[1] ** 2.0 + 182.25 * x[0] * x[1])
class Quintic(Benchmark):
r"""
Quintic objective function.
This class defines the Quintic [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Quintic}}(x) = \sum_{i=1}^{n} \left|{x_{i}^{5} - 3 x_{i}^{4}
+ 4 x_{i}^{3} + 2 x_{i}^{2} - 10 x_{i} -4}\right|
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x_i) = 0` for :math:`x_i = -1` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(-2, 2), (-2, 2)]
self.global_optimum = [[-1.0 for _ in range(self.N)]]
self.fglob = 0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return sum(abs(x ** 5 - 3 * x ** 4 + 4 * x ** 3 + 2 * x ** 2
- 10 * x - 4))

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# -*- coding: utf-8 -*-
from numpy import abs, sum, sin, cos, asarray, arange, pi, exp, log, sqrt
from scipy.optimize import rosen
from .go_benchmark import Benchmark
class Rana(Benchmark):
r"""
Rana objective function.
This class defines the Rana [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Rana}}(x) = \sum_{i=1}^{n} \left[x_{i}
\sin\left(\sqrt{\lvert{x_{1} - x_{i} + 1}\rvert}\right)
\cos\left(\sqrt{\lvert{x_{1} + x_{i} + 1}\rvert}\right) +
\left(x_{1} + 1\right) \sin\left(\sqrt{\lvert{x_{1} + x_{i} +
1}\rvert}\right) \cos\left(\sqrt{\lvert{x_{1} - x_{i} +
1}\rvert}\right)\right]
Here, :math:`n` represents the number of dimensions and :math:`x_i \in
[-500.0, 500.0]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x_i) = -928.5478` for
:math:`x = [-300.3376, 500]`.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: homemade global minimum here.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-500.000001] * self.N,
[500.000001] * self.N))
self.global_optimum = [[-300.3376, 500.]]
self.fglob = -500.8021602966615
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
t1 = sqrt(abs(x[1:] + x[: -1] + 1))
t2 = sqrt(abs(x[1:] - x[: -1] + 1))
v = (x[1:] + 1) * cos(t2) * sin(t1) + x[:-1] * cos(t1) * sin(t2)
return sum(v)
class Rastrigin(Benchmark):
r"""
Rastrigin objective function.
This class defines the Rastrigin [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Rastrigin}}(x) = 10n \sum_{i=1}^n \left[ x_i^2
- 10 \cos(2\pi x_i) \right]
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-5.12, 5.12]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.12] * self.N, [5.12] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return 10.0 * self.N + sum(x ** 2.0 - 10.0 * cos(2.0 * pi * x))
class Ratkowsky01(Benchmark):
"""
Ratkowsky objective function.
.. [1] https://www.itl.nist.gov/div898/strd/nls/data/ratkowsky3.shtml
"""
# TODO, this is a NIST regression standard dataset
def __init__(self, dimensions=4):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0., 1., 0., 0.1],
[1000, 20., 3., 6.]))
self.global_optimum = [[6.996415127e2, 5.2771253025, 7.5962938329e-1,
1.2792483859]]
self.fglob = 8.786404908e3
self.a = asarray([16.08, 33.83, 65.80, 97.20, 191.55, 326.20, 386.87,
520.53, 590.03, 651.92, 724.93, 699.56, 689.96,
637.56, 717.41])
self.b = arange(1, 16.)
def fun(self, x, *args):
self.nfev += 1
vec = x[0] / ((1 + exp(x[1] - x[2] * self.b)) ** (1 / x[3]))
return sum((self.a - vec) ** 2)
class Ratkowsky02(Benchmark):
r"""
Ratkowsky02 objective function.
This class defines the Ratkowsky 2 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Ratkowsky02}}(x) = \sum_{m=1}^{9}(a_m - x[0] / (1 + exp(x[1]
- b_m x[2]))^2
where
.. math::
\begin{cases}
a=[8.93, 10.8, 18.59, 22.33, 39.35, 56.11, 61.73, 64.62, 67.08]\\
b=[9., 14., 21., 28., 42., 57., 63., 70., 79.]\\
\end{cases}
Here :math:`x_1 \in [1, 100]`, :math:`x_2 \in [0.1, 5]` and
:math:`x_3 \in [0.01, 0.5]`
*Global optimum*: :math:`f(x) = 8.0565229338` for
:math:`x = [7.2462237576e1, 2.6180768402, 6.7359200066e-2]`
.. [1] https://www.itl.nist.gov/div898/strd/nls/data/ratkowsky2.shtml
"""
def __init__(self, dimensions=3):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([10, 0.5, 0.01],
[200, 5., 0.5]))
self.global_optimum = [[7.2462237576e1, 2.6180768402, 6.7359200066e-2]]
self.fglob = 8.0565229338
self.a = asarray([8.93, 10.8, 18.59, 22.33, 39.35, 56.11, 61.73, 64.62,
67.08])
self.b = asarray([9., 14., 21., 28., 42., 57., 63., 70., 79.])
def fun(self, x, *args):
self.nfev += 1
vec = x[0] / (1 + exp(x[1] - x[2] * self.b))
return sum((self.a - vec) ** 2)
class Ripple01(Benchmark):
r"""
Ripple 1 objective function.
This class defines the Ripple 1 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Ripple01}}(x) = \sum_{i=1}^2 -e^{-2 \log 2
(\frac{x_i-0.1}{0.8})^2} \left[\sin^6(5 \pi x_i)
+ 0.1\cos^2(500 \pi x_i) \right]
with :math:`x_i \in [0, 1]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -2.2` for :math:`x_i = 0.1` for
:math:`i = 1, 2`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [1.0] * self.N))
self.global_optimum = [[0.1 for _ in range(self.N)]]
self.fglob = -2.2
def fun(self, x, *args):
self.nfev += 1
u = -2.0 * log(2.0) * ((x - 0.1) / 0.8) ** 2.0
v = sin(5.0 * pi * x) ** 6.0 + 0.1 * cos(500.0 * pi * x) ** 2.0
return sum(-exp(u) * v)
class Ripple25(Benchmark):
r"""
Ripple 25 objective function.
This class defines the Ripple 25 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Ripple25}}(x) = \sum_{i=1}^2 -e^{-2
\log 2 (\frac{x_i-0.1}{0.8})^2}
\left[\sin^6(5 \pi x_i) \right]
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [0, 1]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = -2` for :math:`x_i = 0.1` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [1.0] * self.N))
self.global_optimum = [[0.1 for _ in range(self.N)]]
self.fglob = -2.0
def fun(self, x, *args):
self.nfev += 1
u = -2.0 * log(2.0) * ((x - 0.1) / 0.8) ** 2.0
v = sin(5.0 * pi * x) ** 6.0
return sum(-exp(u) * v)
class Rosenbrock(Benchmark):
r"""
Rosenbrock objective function.
This class defines the Rosenbrock [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Rosenbrock}}(x) = \sum_{i=1}^{n-1} [100(x_i^2
- x_{i+1})^2 + (x_i - 1)^2]
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-5, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 1` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-30.] * self.N, [30.0] * self.N))
self.custom_bounds = [(-2, 2), (-2, 2)]
self.global_optimum = [[1 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return rosen(x)
class RosenbrockModified(Benchmark):
r"""
Modified Rosenbrock objective function.
This class defines the Modified Rosenbrock [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{RosenbrockModified}}(x) = 74 + 100(x_2 - x_1^2)^2
+ (1 - x_1)^2 - 400 e^{-\frac{(x_1+1)^2 + (x_2 + 1)^2}{0.1}}
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-2, 2]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 34.04024310` for
:math:`x = [-0.90955374, -0.95057172]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: We have different global minimum compared to Jamil #106. This is
possibly because of the (1-x) term is using the wrong parameter.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-2.0] * self.N, [2.0] * self.N))
self.custom_bounds = ([-1.0, 0.5], [-1.0, 1.0])
self.global_optimum = [[-0.90955374, -0.95057172]]
self.fglob = 34.040243106640844
def fun(self, x, *args):
self.nfev += 1
a = 74 + 100. * (x[1] - x[0] ** 2) ** 2 + (1 - x[0]) ** 2
a -= 400 * exp(-((x[0] + 1.) ** 2 + (x[1] + 1.) ** 2) / 0.1)
return a
class RotatedEllipse01(Benchmark):
r"""
Rotated Ellipse 1 objective function.
This class defines the Rotated Ellipse 1 [1]_ global optimization problem. This
is a unimodal minimization problem defined as follows:
.. math::
f_{\text{RotatedEllipse01}}(x) = 7x_1^2 - 6 \sqrt{3} x_1x_2 + 13x_2^2
with :math:`x_i \in [-500, 500]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-500.0] * self.N,
[500.0] * self.N))
self.custom_bounds = ([-2.0, 2.0], [-2.0, 2.0])
self.global_optimum = [[0.0, 0.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return (7.0 * x[0] ** 2.0 - 6.0 * sqrt(3) * x[0] * x[1]
+ 13 * x[1] ** 2.0)
class RotatedEllipse02(Benchmark):
r"""
Rotated Ellipse 2 objective function.
This class defines the Rotated Ellipse 2 [1]_ global optimization problem. This
is a unimodal minimization problem defined as follows:
.. math::
f_{\text{RotatedEllipse02}}(x) = x_1^2 - x_1 x_2 + x_2^2
with :math:`x_i \in [-500, 500]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-500.0] * self.N,
[500.0] * self.N))
self.custom_bounds = ([-2.0, 2.0], [-2.0, 2.0])
self.global_optimum = [[0.0, 0.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return x[0] ** 2.0 - x[0] * x[1] + x[1] ** 2.0

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# -*- coding: utf-8 -*-
from numpy import abs, asarray, cos, exp, arange, pi, sin, sum, atleast_2d
from .go_benchmark import Benchmark
class TestTubeHolder(Benchmark):
r"""
TestTubeHolder objective function.
This class defines the TestTubeHolder [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{TestTubeHolder}}(x) = - 4 \left | {e^{\left|{\cos
\left(\frac{1}{200} x_{1}^{2} + \frac{1}{200} x_{2}^{2}\right)}
\right|}\sin\left(x_{1}\right) \cos\left(x_{2}\right)}\right|
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -10.872299901558` for
:math:`x= [-\pi/2, 0]`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO Jamil#148 has got incorrect equation, missing an abs around the square
brackets
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[-pi / 2, 0.0]]
self.fglob = -10.87229990155800
def fun(self, x, *args):
self.nfev += 1
u = sin(x[0]) * cos(x[1])
v = (x[0] ** 2 + x[1] ** 2) / 200
return -4 * abs(u * exp(abs(cos(v))))
class Thurber(Benchmark):
r"""
Thurber [1]_ objective function.
.. [1] https://www.itl.nist.gov/div898/strd/nls/data/thurber.shtml
"""
def __init__(self, dimensions=7):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip(
[500., 500., 100., 10., 0.1, 0.1, 0.],
[2000., 2000., 1000., 150., 2., 1., 0.2]))
self.global_optimum = [[1.288139680e3, 1.4910792535e3, 5.8323836877e2,
75.416644291, 0.96629502864, 0.39797285797,
4.9727297349e-2]]
self.fglob = 5642.7082397
self.a = asarray([80.574, 84.248, 87.264, 87.195, 89.076, 89.608,
89.868, 90.101, 92.405, 95.854, 100.696, 101.06,
401.672, 390.724, 567.534, 635.316, 733.054, 759.087,
894.206, 990.785, 1090.109, 1080.914, 1122.643,
1178.351, 1260.531, 1273.514, 1288.339, 1327.543,
1353.863, 1414.509, 1425.208, 1421.384, 1442.962,
1464.350, 1468.705, 1447.894, 1457.628])
self.b = asarray([-3.067, -2.981, -2.921, -2.912, -2.840, -2.797,
-2.702, -2.699, -2.633, -2.481, -2.363, -2.322,
-1.501, -1.460, -1.274, -1.212, -1.100, -1.046,
-0.915, -0.714, -0.566, -0.545, -0.400, -0.309,
-0.109, -0.103, 0.010, 0.119, 0.377, 0.790, 0.963,
1.006, 1.115, 1.572, 1.841, 2.047, 2.200])
def fun(self, x, *args):
self.nfev += 1
vec = x[0] + x[1] * self.b + x[2] * self.b ** 2 + x[3] * self.b ** 3
vec /= 1 + x[4] * self.b + x[5] * self.b ** 2 + x[6] * self.b ** 3
return sum((self.a - vec) ** 2)
class Treccani(Benchmark):
r"""
Treccani objective function.
This class defines the Treccani [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Treccani}}(x) = x_1^4 + 4x_1^3 + 4x_1^2 + x_2^2
with :math:`x_i \in
[-5, 5]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [-2, 0]` or
:math:`x = [0, 0]`.
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
self.custom_bounds = [(-2, 2), (-2, 2)]
self.global_optimum = [[-2.0, 0.0]]
self.fglob = 0
def fun(self, x, *args):
self.nfev += 1
return x[0] ** 4 + 4.0 * x[0] ** 3 + 4.0 * x[0] ** 2 + x[1] ** 2
class Trefethen(Benchmark):
r"""
Trefethen objective function.
This class defines the Trefethen [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Trefethen}}(x) = 0.25 x_{1}^{2} + 0.25 x_{2}^{2}
+ e^{\sin\left(50 x_{1}\right)}
- \sin\left(10 x_{1} + 10 x_{2}\right)
+ \sin\left(60 e^{x_{2}}\right)
+ \sin\left[70 \sin\left(x_{1}\right)\right]
+ \sin\left[\sin\left(80 x_{2}\right)\right]
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -3.3068686474` for
:math:`x = [-0.02440307923, 0.2106124261]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = [(-5, 5), (-5, 5)]
self.global_optimum = [[-0.02440307923, 0.2106124261]]
self.fglob = -3.3068686474
def fun(self, x, *args):
self.nfev += 1
val = 0.25 * x[0] ** 2 + 0.25 * x[1] ** 2
val += exp(sin(50. * x[0])) - sin(10 * x[0] + 10 * x[1])
val += sin(60 * exp(x[1]))
val += sin(70 * sin(x[0]))
val += sin(sin(80 * x[1]))
return val
class ThreeHumpCamel(Benchmark):
r"""
Three Hump Camel objective function.
This class defines the Three Hump Camel [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{ThreeHumpCamel}}(x) = 2x_1^2 - 1.05x_1^4 + \frac{x_1^6}{6}
+ x_1x_2 + x_2^2
with :math:`x_i \in [-5, 5]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
self.custom_bounds = [(-2, 2), (-1.5, 1.5)]
self.global_optimum = [[0.0, 0.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return (2.0 * x[0] ** 2.0 - 1.05 * x[0] ** 4.0 + x[0] ** 6 / 6.0
+ x[0] * x[1] + x[1] ** 2.0)
class Trid(Benchmark):
r"""
Trid objective function.
This class defines the Trid [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Trid}}(x) = \sum_{i=1}^{n} (x_i - 1)^2
- \sum_{i=2}^{n} x_i x_{i-1}
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-20, 20]` for :math:`i = 1, ..., 6`.
*Global optimum*: :math:`f(x) = -50` for :math:`x = [6, 10, 12, 12, 10, 6]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO Jamil#150, starting index of second summation term should be 2.
"""
def __init__(self, dimensions=6):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-20.0] * self.N, [20.0] * self.N))
self.global_optimum = [[6, 10, 12, 12, 10, 6]]
self.fglob = -50.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return sum((x - 1.0) ** 2.0) - sum(x[1:] * x[:-1])
class Trigonometric01(Benchmark):
r"""
Trigonometric 1 objective function.
This class defines the Trigonometric 1 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Trigonometric01}}(x) = \sum_{i=1}^{n} \left [n -
\sum_{j=1}^{n} \cos(x_j)
+ i \left(1 - cos(x_i)
- sin(x_i) \right ) \right]^2
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [0, \pi]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO: equaiton uncertain here. Is it just supposed to be the cos term
in the inner sum, or the whole of the second line in Jamil #153.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [pi] * self.N))
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
i = atleast_2d(arange(1.0, self.N + 1)).T
inner = cos(x) + i * (1 - cos(x) - sin(x))
return sum((self.N - sum(inner, axis=1)) ** 2)
class Trigonometric02(Benchmark):
r"""
Trigonometric 2 objective function.
This class defines the Trigonometric 2 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Trigonometric2}}(x) = 1 + \sum_{i=1}^{n} 8 \sin^2
\left[7(x_i - 0.9)^2 \right]
+ 6 \sin^2 \left[14(x_i - 0.9)^2 \right]
+ (x_i - 0.9)^2
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-500, 500]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 1` for :math:`x_i = 0.9` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-500.0] * self.N,
[500.0] * self.N))
self.custom_bounds = [(0, 2), (0, 2)]
self.global_optimum = [[0.9 for _ in range(self.N)]]
self.fglob = 1.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
vec = (8 * sin(7 * (x - 0.9) ** 2) ** 2
+ 6 * sin(14 * (x - 0.9) ** 2) ** 2
+ (x - 0.9) ** 2)
return 1.0 + sum(vec)
class Tripod(Benchmark):
r"""
Tripod objective function.
This class defines the Tripod [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Tripod}}(x) = p(x_2) \left[1 + p(x_1) \right] +
\lvert x_1 + 50p(x_2) \left[1 - 2p(x_1) \right]
\rvert + \lvert x_2 + 50\left[1 - 2p(x_2)\right]
\rvert
with :math:`x_i \in [-100, 100]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, -50]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-100.0] * self.N,
[100.0] * self.N))
self.global_optimum = [[0.0, -50.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
p1 = float(x[0] >= 0)
p2 = float(x[1] >= 0)
return (p2 * (1.0 + p1) + abs(x[0] + 50.0 * p2 * (1.0 - 2.0 * p1))
+ abs(x[1] + 50.0 * (1.0 - 2.0 * p2)))

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# -*- coding: utf-8 -*-
from numpy import abs, sin, cos, pi, sqrt
from .go_benchmark import Benchmark
class Ursem01(Benchmark):
r"""
Ursem 1 objective function.
This class defines the Ursem 1 [1]_ global optimization problem. This is a
unimodal minimization problem defined as follows:
.. math::
f_{\text{Ursem01}}(x) = - \sin(2x_1 - 0.5 \pi) - 3 \cos(x_2) - 0.5 x_1
with :math:`x_1 \in [-2.5, 3]` and :math:`x_2 \in [-2, 2]`.
*Global optimum*: :math:`f(x) = -4.81681406371` for
:math:`x = [1.69714, 0.0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = [(-2.5, 3.0), (-2.0, 2.0)]
self.global_optimum = [[1.69714, 0.0]]
self.fglob = -4.81681406371
def fun(self, x, *args):
self.nfev += 1
return -sin(2 * x[0] - 0.5 * pi) - 3.0 * cos(x[1]) - 0.5 * x[0]
class Ursem03(Benchmark):
r"""
Ursem 3 objective function.
This class defines the Ursem 3 [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Ursem03}}(x) = - \sin(2.2 \pi x_1 + 0.5 \pi)
\frac{2 - \lvert x_1 \rvert}{2}
\frac{3 - \lvert x_1 \rvert}{2}
- \sin(2.2 \pi x_2 + 0.5 \pi)
\frac{2 - \lvert x_2 \rvert}{2}
\frac{3 - \lvert x_2 \rvert}{2}
with :math:`x_1 \in [-2, 2]`, :math:`x_2 \in [-1.5, 1.5]`.
*Global optimum*: :math:`f(x) = -3` for :math:`x = [0, 0]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO Gavana and Jamil #157 disagree on the formulae here. Jamil squares the
x[1] term in the sine expression. Gavana doesn't. Go with Gavana here.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = [(-2, 2), (-1.5, 1.5)]
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = -3.0
def fun(self, x, *args):
self.nfev += 1
u = -(sin(2.2 * pi * x[0] + 0.5 * pi)
* ((2.0 - abs(x[0])) / 2.0) * ((3.0 - abs(x[0])) / 2))
v = -(sin(2.2 * pi * x[1] + 0.5 * pi)
* ((2.0 - abs(x[1])) / 2) * ((3.0 - abs(x[1])) / 2))
return u + v
class Ursem04(Benchmark):
r"""
Ursem 4 objective function.
This class defines the Ursem 4 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Ursem04}}(x) = -3 \sin(0.5 \pi x_1 + 0.5 \pi)
\frac{2 - \sqrt{x_1^2 + x_2 ^ 2}}{4}
with :math:`x_i \in [-2, 2]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -1.5` for :math:`x = [0, 0]` for
:math:`i = 1, 2`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-2.0] * self.N, [2.0] * self.N))
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = -1.5
def fun(self, x, *args):
self.nfev += 1
return (-3 * sin(0.5 * pi * x[0] + 0.5 * pi)
* (2 - sqrt(x[0] ** 2 + x[1] ** 2)) / 4)
class UrsemWaves(Benchmark):
r"""
Ursem Waves objective function.
This class defines the Ursem Waves [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{UrsemWaves}}(x) = -0.9x_1^2 + (x_2^2 - 4.5x_2^2)x_1x_2
+ 4.7 \cos \left[ 2x_1 - x_2^2(2 + x_1)
\right ] \sin(2.5 \pi x_1)
with :math:`x_1 \in [-0.9, 1.2]`, :math:`x_2 \in [-1.2, 1.2]`.
*Global optimum*: :math:`f(x) = -8.5536` for :math:`x = [1.2, 1.2]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO Jamil #159, has an x_2^2 - 4.5 x_2^2 in the brackets. Why wasn't this
rationalised to -5.5 x_2^2? This makes me wonder if the equation is listed
correctly?
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = [(-0.9, 1.2), (-1.2, 1.2)]
self.global_optimum = [[1.2 for _ in range(self.N)]]
self.fglob = -8.5536
def fun(self, x, *args):
self.nfev += 1
u = -0.9 * x[0] ** 2
v = (x[1] ** 2 - 4.5 * x[1] ** 2) * x[0] * x[1]
w = 4.7 * cos(3 * x[0] - x[1] ** 2 * (2 + x[0])) * sin(2.5 * pi * x[0])
return u + v + w

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# -*- coding: utf-8 -*-
from numpy import sum, cos, sin, log
from .go_benchmark import Benchmark
class VenterSobiezcczanskiSobieski(Benchmark):
r"""
Venter Sobiezcczanski-Sobieski objective function.
This class defines the Venter Sobiezcczanski-Sobieski [1]_ global optimization
problem. This is a multimodal minimization problem defined as follows:
.. math::
f_{\text{VenterSobiezcczanskiSobieski}}(x) = x_1^2 - 100 \cos^2(x_1)
- 100 \cos(x_1^2/30)
+ x_2^2 - 100 \cos^2(x_2)
- 100 \cos(x_2^2/30)
with :math:`x_i \in [-50, 50]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -400` for :math:`x = [0, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO Jamil #160 hasn't written the equation very well. Normally a cos
squared term is written as cos^2(x) rather than cos(x)^2
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-50.0] * self.N, [50.0] * self.N))
self.custom_bounds = ([-10, 10], [-10, 10])
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = -400
def fun(self, x, *args):
self.nfev += 1
u = x[0] ** 2.0 - 100.0 * cos(x[0]) ** 2.0
v = -100.0 * cos(x[0] ** 2.0 / 30.0) + x[1] ** 2.0
w = - 100.0 * cos(x[1]) ** 2.0 - 100.0 * cos(x[1] ** 2.0 / 30.0)
return u + v + w
class Vincent(Benchmark):
r"""
Vincent objective function.
This class defines the Vincent [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Vincent}}(x) = - \sum_{i=1}^{n} \sin(10 \log(x))
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [0.25, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = -n` for :math:`x_i = 7.70628098`
for :math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.25] * self.N, [10.0] * self.N))
self.global_optimum = [[7.70628098 for _ in range(self.N)]]
self.fglob = -float(self.N)
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return -sum(sin(10.0 * log(x)))

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# -*- coding: utf-8 -*-
from numpy import atleast_2d, arange, sum, cos, exp, pi
from .go_benchmark import Benchmark
class Watson(Benchmark):
r"""
Watson objective function.
This class defines the Watson [1]_ global optimization problem. This is a
unimodal minimization problem defined as follows:
.. math::
f_{\text{Watson}}(x) = \sum_{i=0}^{29} \left\{
\sum_{j=0}^4 ((j + 1)a_i^j x_{j+1})
- \left[ \sum_{j=0}^5 a_i^j
x_{j+1} \right ]^2 - 1 \right\}^2
+ x_1^2
Where, in this exercise, :math:`a_i = i/29`.
with :math:`x_i \in [-5, 5]` for :math:`i = 1, ..., 6`.
*Global optimum*: :math:`f(x) = 0.002288` for
:math:`x = [-0.0158, 1.012, -0.2329, 1.260, -1.513, 0.9928]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
TODO Jamil #161 writes equation using (j - 1). According to code in Adorio
and Gavana it should be (j+1). However the equations in those papers
contain (j - 1) as well. However, I've got the right global minimum!!!
"""
def __init__(self, dimensions=6):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
self.global_optimum = [[-0.0158, 1.012, -0.2329, 1.260, -1.513,
0.9928]]
self.fglob = 0.002288
def fun(self, x, *args):
self.nfev += 1
i = atleast_2d(arange(30.)).T
a = i / 29.
j = arange(5.)
k = arange(6.)
t1 = sum((j + 1) * a ** j * x[1:], axis=1)
t2 = sum(a ** k * x, axis=1)
inner = (t1 - t2 ** 2 - 1) ** 2
return sum(inner) + x[0] ** 2
class Wavy(Benchmark):
r"""
Wavy objective function.
This class defines the W / Wavy [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Wavy}}(x) = 1 - \frac{1}{n} \sum_{i=1}^{n}
\cos(kx_i)e^{-\frac{x_i^2}{2}}
Where, in this exercise, :math:`k = 10`. The number of local minima is
:math:`kn` and :math:`(k + 1)n` for odd and even :math:`k` respectively.
Here, :math:`x_i \in [-\pi, \pi]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-pi] * self.N, [pi] * self.N))
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return 1.0 - (1.0 / self.N) * sum(cos(10 * x) * exp(-x ** 2.0 / 2.0))
class WayburnSeader01(Benchmark):
r"""
Wayburn and Seader 1 objective function.
This class defines the Wayburn and Seader 1 [1]_ global optimization
problem. This is a unimodal minimization problem defined as follows:
.. math::
f_{\text{WayburnSeader01}}(x) = (x_1^6 + x_2^4 - 17)^2
+ (2x_1 + x_2 - 4)^2
with :math:`x_i \in [-5, 5]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 2]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
self.custom_bounds = ([-2, 2], [-2, 2])
self.global_optimum = [[1.0, 2.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return (x[0] ** 6 + x[1] ** 4 - 17) ** 2 + (2 * x[0] + x[1] - 4) ** 2
class WayburnSeader02(Benchmark):
r"""
Wayburn and Seader 2 objective function.
This class defines the Wayburn and Seader 2 [1]_ global optimization
problem. This is a unimodal minimization problem defined as follows:
.. math::
f_{\text{WayburnSeader02}}(x) = \left[ 1.613 - 4(x_1 - 0.3125)^2
- 4(x_2 - 1.625)^2 \right]^2
+ (x_2 - 1)^2
with :math:`x_i \in [-500, 500]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0.2, 1]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-500.0] * self.N,
[500.0] * self.N))
self.custom_bounds = ([-1, 2], [-1, 2])
self.global_optimum = [[0.2, 1.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
u = (1.613 - 4 * (x[0] - 0.3125) ** 2 - 4 * (x[1] - 1.625) ** 2) ** 2
v = (x[1] - 1) ** 2
return u + v
class Weierstrass(Benchmark):
r"""
Weierstrass objective function.
This class defines the Weierstrass [1]_ global optimization problem.
This is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Weierstrass}}(x) = \sum_{i=1}^{n} \left [
\sum_{k=0}^{kmax} a^k \cos
\left( 2 \pi b^k (x_i + 0.5) \right) - n
\sum_{k=0}^{kmax} a^k \cos(\pi b^k) \right ]
Where, in this exercise, :math:`kmax = 20`, :math:`a = 0.5` and
:math:`b = 3`.
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-0.5, 0.5]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 4` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO line 1591.
TODO Jamil, Gavana have got it wrong. The second term is not supposed to
be included in the outer sum. Mishra code has it right as does the
reference referred to in Jamil#166.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-0.5] * self.N, [0.5] * self.N))
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = 0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
kmax = 20
a, b = 0.5, 3.0
k = atleast_2d(arange(kmax + 1.)).T
t1 = a ** k * cos(2 * pi * b ** k * (x + 0.5))
t2 = self.N * sum(a ** k.T * cos(pi * b ** k.T))
return sum(sum(t1, axis=0)) - t2
class Whitley(Benchmark):
r"""
Whitley objective function.
This class defines the Whitley [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Whitley}}(x) = \sum_{i=1}^n \sum_{j=1}^n
\left[\frac{(100(x_i^2-x_j)^2
+ (1-x_j)^2)^2}{4000} - \cos(100(x_i^2-x_j)^2
+ (1-x_j)^2)+1 \right]
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10.24, 10.24]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 1` for
:math:`i = 1, ..., n`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO Jamil#167 has '+ 1' inside the cos term, when it should be outside it.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.24] * self.N,
[10.24] * self.N))
self.custom_bounds = ([-1, 2], [-1, 2])
self.global_optimum = [[1.0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
XI = x
XJ = atleast_2d(x).T
temp = 100.0 * ((XI ** 2.0) - XJ) + (1.0 - XJ) ** 2.0
inner = (temp ** 2.0 / 4000.0) - cos(temp) + 1.0
return sum(sum(inner, axis=0))
class Wolfe(Benchmark):
r"""
Wolfe objective function.
This class defines the Wolfe [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Wolfe}}(x) = \frac{4}{3}(x_1^2 + x_2^2 - x_1x_2)^{0.75} + x_3
with :math:`x_i \in [0, 2]` for :math:`i = 1, 2, 3`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, 0, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=3):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [2.0] * self.N))
self.global_optimum = [[0.0 for _ in range(self.N)]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
return 4 / 3 * (x[0] ** 2 + x[1] ** 2 - x[0] * x[1]) ** 0.75 + x[2]

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# -*- coding: utf-8 -*-
import numpy as np
from numpy import abs, sum, sin, cos, pi, exp, arange, prod, sqrt
from .go_benchmark import Benchmark
class XinSheYang01(Benchmark):
r"""
Xin-She Yang 1 objective function.
This class defines the Xin-She Yang 1 [1]_ global optimization problem.
This is a multimodal minimization problem defined as follows:
.. math::
f_{\text{XinSheYang01}}(x) = \sum_{i=1}^{n} \epsilon_i \lvert x_i
\rvert^i
The variable :math:`\epsilon_i, (i = 1, ..., n)` is a random variable
uniformly distributed in :math:`[0, 1]`.
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-5, 5]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
self.custom_bounds = ([-2, 2], [-2, 2])
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
i = arange(1.0, self.N + 1.0)
return sum(np.random.random(self.N) * (abs(x) ** i))
class XinSheYang02(Benchmark):
r"""
Xin-She Yang 2 objective function.
This class defines the Xin-She Yang 2 [1]_ global optimization problem.
This is a multimodal minimization problem defined as follows:
.. math::
f_{\text{XinSheYang02}}(\x) = \frac{\sum_{i=1}^{n} \lvert{x_{i}}\rvert}
{e^{\sum_{i=1}^{n} \sin\left(x_{i}^{2.0}
\right)}}
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-2\pi, 2\pi]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-2 * pi] * self.N,
[2 * pi] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return sum(abs(x)) * exp(-sum(sin(x ** 2.0)))
class XinSheYang03(Benchmark):
r"""
Xin-She Yang 3 objective function.
This class defines the Xin-She Yang 3 [1]_ global optimization problem.
This is a multimodal minimization problem defined as follows:
.. math::
f_{\text{XinSheYang03}}(x) = e^{-\sum_{i=1}^{n} (x_i/\beta)^{2m}}
- 2e^{-\sum_{i=1}^{n} x_i^2}
\prod_{i=1}^{n} \cos^2(x_i)
Where, in this exercise, :math:`\beta = 15` and :math:`m = 3`.
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-20, 20]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = -1` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-20.0] * self.N, [20.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = -1.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
beta, m = 15.0, 5.0
u = sum((x / beta) ** (2 * m))
v = sum(x ** 2)
w = prod(cos(x) ** 2)
return exp(-u) - 2 * exp(-v) * w
class XinSheYang04(Benchmark):
r"""
Xin-She Yang 4 objective function.
This class defines the Xin-She Yang 4 [1]_ global optimization problem.
This is a multimodal minimization problem defined as follows:
.. math::
f_{\text{XinSheYang04}}(x) = \left[ \sum_{i=1}^{n} \sin^2(x_i)
- e^{-\sum_{i=1}^{n} x_i^2} \right ]
e^{-\sum_{i=1}^{n} \sin^2 \sqrt{ \lvert
x_i \rvert }}
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = -1` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = -1.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
u = sum(sin(x) ** 2)
v = sum(x ** 2)
w = sum(sin(sqrt(abs(x))) ** 2)
return (u - exp(-v)) * exp(-w)
class Xor(Benchmark):
r"""
Xor objective function.
This class defines the Xor [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Xor}}(x) = \left[ 1 + \exp \left( - \frac{x_7}{1 +
\exp(-x_1 - x_2 - x_5)} - \frac{x_8}{1 + \exp(-x_3 - x_4 - x_6)}
- x_9 \right ) \right ]^{-2} \\
+ \left [ 1 + \exp \left( -\frac{x_7}{1 + \exp(-x_5)}
- \frac{x_8}{1 + \exp(-x_6)} - x_9 \right ) \right] ^{-2} \\
+ \left [1 - \left\{1 + \exp \left(-\frac{x_7}{1 + \exp(-x_1 - x_5)}
- \frac{x_8}{1 + \exp(-x_3 - x_6)} - x_9 \right ) \right\}^{-1}
\right ]^2 \\
+ \left [1 - \left\{1 + \exp \left(-\frac{x_7}{1 + \exp(-x_2 - x_5)}
- \frac{x_8}{1 + \exp(-x_4 - x_6)} - x_9 \right ) \right\}^{-1}
\right ]^2
with :math:`x_i \in [-1, 1]` for :math:`i=1,...,9`.
*Global optimum*: :math:`f(x) = 0.9597588` for
:math:`\x = [1, -1, 1, -1, -1, 1, 1, -1, 0.421134]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=9):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-1.0] * self.N, [1.0] * self.N))
self.global_optimum = [[1.0, -1.0, 1.0,
-1.0, -1.0, 1.0, 1.0, -1.0, 0.421134]]
self.fglob = 0.9597588
def fun(self, x, *args):
self.nfev += 1
F11 = x[6] / (1.0 + exp(-x[0] - x[1] - x[4]))
F12 = x[7] / (1.0 + exp(-x[2] - x[3] - x[5]))
F1 = (1.0 + exp(-F11 - F12 - x[8])) ** (-2)
F21 = x[6] / (1.0 + exp(-x[4]))
F22 = x[7] / (1.0 + exp(-x[5]))
F2 = (1.0 + exp(-F21 - F22 - x[8])) ** (-2)
F31 = x[6] / (1.0 + exp(-x[0] - x[4]))
F32 = x[7] / (1.0 + exp(-x[2] - x[5]))
F3 = (1.0 - (1.0 + exp(-F31 - F32 - x[8])) ** (-1)) ** 2
F41 = x[6] / (1.0 + exp(-x[1] - x[4]))
F42 = x[7] / (1.0 + exp(-x[3] - x[5]))
F4 = (1.0 - (1.0 + exp(-F41 - F42 - x[8])) ** (-1)) ** 2
return F1 + F2 + F3 + F4

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# -*- coding: utf-8 -*-
from numpy import abs, sum, cos, pi
from .go_benchmark import Benchmark
class YaoLiu04(Benchmark):
r"""
Yao-Liu 4 objective function.
This class defines the Yao-Liu function 4 [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{YaoLiu04}}(x) = {max}_i \left\{ \left | x_i \right | ,
1 \leq i \leq n \right\}
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Yao X., Liu Y. (1997) Fast evolution strategies.
In: Angeline P.J., Reynolds R.G., McDonnell J.R., Eberhart R. (eds)
Evolutionary Programming VI. EP 1997.
Lecture Notes in Computer Science, vol 1213. Springer, Berlin, Heidelberg
.. [2] Mishra, S. Global Optimization by Differential Evolution and
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
Munich Personal RePEc Archive, 2006, 1005
TODO line 1201. Gavana code and documentation differ.
max(abs(x)) != abs(max(x))
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return abs(x).max()
class YaoLiu09(Benchmark):
r"""
Yao-Liu 9 objective function.
This class defines the Yao-Liu [1]_ function 9 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{YaoLiu09}}(x) = \sum_{i=1}^n \left [ x_i^2
- 10 \cos(2 \pi x_i ) + 10 \right ]
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-5.12, 5.12]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Yao X., Liu Y. (1997) Fast evolution strategies.
In: Angeline P.J., Reynolds R.G., McDonnell J.R., Eberhart R. (eds)
Evolutionary Programming VI. EP 1997.
Lecture Notes in Computer Science, vol 1213. Springer, Berlin, Heidelberg
.. [2] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.12] * self.N, [5.12] * self.N))
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
return sum(x ** 2.0 - 10.0 * cos(2 * pi * x) + 10)

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# -*- coding: utf-8 -*-
from numpy import abs, sum, sign, arange
from .go_benchmark import Benchmark
class Zacharov(Benchmark):
r"""
Zacharov objective function.
This class defines the Zacharov [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Zacharov}}(x) = \sum_{i=1}^{n} x_i^2 + \left ( \frac{1}{2}
\sum_{i=1}^{n} i x_i \right )^2
+ \left ( \frac{1}{2} \sum_{i=1}^{n} i x_i
\right )^4
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-5, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
:math:`i = 1, ..., n`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [10.0] * self.N))
self.custom_bounds = ([-1, 1], [-1, 1])
self.global_optimum = [[0 for _ in range(self.N)]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
u = sum(x ** 2)
v = sum(arange(1, self.N + 1) * x)
return u + (0.5 * v) ** 2 + (0.5 * v) ** 4
class ZeroSum(Benchmark):
r"""
ZeroSum objective function.
This class defines the ZeroSum [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{ZeroSum}}(x) = \begin{cases}
0 & \textrm{if} \sum_{i=1}^n x_i = 0 \\
1 + \left(10000 \left |\sum_{i=1}^n x_i\right|
\right)^{0.5} & \textrm{otherwise}
\end{cases}
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0` where :math:`\sum_{i=1}^n x_i = 0`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.global_optimum = [[]]
self.fglob = 0.0
self.change_dimensionality = True
def fun(self, x, *args):
self.nfev += 1
if abs(sum(x)) < 3e-16:
return 0.0
return 1.0 + (10000.0 * abs(sum(x))) ** 0.5
class Zettl(Benchmark):
r"""
Zettl objective function.
This class defines the Zettl [1]_ global optimization problem. This is a
multimodal minimization problem defined as follows:
.. math::
f_{\text{Zettl}}(x) = \frac{1}{4} x_{1} + \left(x_{1}^{2} - 2 x_{1}
+ x_{2}^{2}\right)^{2}
with :math:`x_i \in [-1, 5]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -0.0037912` for :math:`x = [-0.029896, 0.0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-5.0] * self.N, [10.0] * self.N))
self.global_optimum = [[-0.02989597760285287, 0.0]]
self.fglob = -0.003791237220468656
def fun(self, x, *args):
self.nfev += 1
return (x[0] ** 2 + x[1] ** 2 - 2 * x[0]) ** 2 + 0.25 * x[0]
class Zimmerman(Benchmark):
r"""
Zimmerman objective function.
This class defines the Zimmerman [1]_ global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\text{Zimmerman}}(x) = \max \left[Zh1(x), Zp(Zh2(x))
\textrm{sgn}(Zh2(x)), Zp(Zh3(x))
\textrm{sgn}(Zh3(x)),
Zp(-x_1)\textrm{sgn}(x_1),
Zp(-x_2)\textrm{sgn}(x_2) \right]
Where, in this exercise:
.. math::
\begin{cases}
Zh1(x) = 9 - x_1 - x_2 \\
Zh2(x) = (x_1 - 3)^2 + (x_2 - 2)^2 \\
Zh3(x) = x_1x_2 - 14 \\
Zp(t) = 100(1 + t)
\end{cases}
Where :math:`x` is a vector and :math:`t` is a scalar.
Here, :math:`x_i \in [0, 100]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [7, 2]`
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
TODO implementation from Gavana
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([0.0] * self.N, [100.0] * self.N))
self.custom_bounds = ([0.0, 8.0], [0.0, 8.0])
self.global_optimum = [[7.0, 2.0]]
self.fglob = 0.0
def fun(self, x, *args):
self.nfev += 1
Zh1 = lambda x: 9.0 - x[0] - x[1]
Zh2 = lambda x: (x[0] - 3.0) ** 2.0 + (x[1] - 2.0) ** 2.0 - 16.0
Zh3 = lambda x: x[0] * x[1] - 14.0
Zp = lambda x: 100.0 * (1.0 + x)
return max(Zh1(x),
Zp(Zh2(x)) * sign(Zh2(x)),
Zp(Zh3(x)) * sign(Zh3(x)),
Zp(-x[0]) * sign(x[0]),
Zp(-x[1]) * sign(x[1]))
class Zirilli(Benchmark):
r"""
Zettl objective function.
This class defines the Zirilli [1]_ global optimization problem. This is a
unimodal minimization problem defined as follows:
.. math::
f_{\text{Zirilli}}(x) = 0.25x_1^4 - 0.5x_1^2 + 0.1x_1 + 0.5x_2^2
Here, :math:`n` represents the number of dimensions and
:math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -0.3523` for :math:`x = [-1.0465, 0]`
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
For Global Optimization Problems Int. Journal of Mathematical Modelling
and Numerical Optimisation, 2013, 4, 150-194.
"""
def __init__(self, dimensions=2):
Benchmark.__init__(self, dimensions)
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
self.custom_bounds = ([-2.0, 2.0], [-2.0, 2.0])
self.global_optimum = [[-1.0465, 0.0]]
self.fglob = -0.35238603
def fun(self, x, *args):
self.nfev += 1
return 0.25 * x[0] ** 4 - 0.5 * x[0] ** 2 + 0.1 * x[0] + 0.5 * x[1] ** 2

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# -*- coding: utf-8 -*-
from numpy import cos, exp, log, pi, sin, sqrt
from .go_benchmark import Benchmark, safe_import
with safe_import():
try:
from scipy.special import factorial # new
except ImportError:
from scipy.misc import factorial # old
#-----------------------------------------------------------------------
# UNIVARIATE SINGLE-OBJECTIVE PROBLEMS
#-----------------------------------------------------------------------
class Problem02(Benchmark):
"""
Univariate Problem02 objective function.
This class defines the Univariate Problem02 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem02}}(x) = \\sin(x) + \\sin \\left(\\frac{10}{3}x \\right)
Bound constraints: :math:`x \\in [2.7, 7.5]`
.. figure:: figures/Problem02.png
:alt: Univariate Problem02 function
:align: center
**Univariate Problem02 function**
*Global optimum*: :math:`f(x)=-1.899599` for :math:`x = 5.145735`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(2.7, 7.5)]
self.global_optimum = 5.145735
self.fglob = -1.899599
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return sin(x) + sin(10.0 / 3.0 * x)
class Problem03(Benchmark):
"""
Univariate Problem03 objective function.
This class defines the Univariate Problem03 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem03}}(x) = - \\sum_{k=1}^6 k \\sin[(k+1)x+k]
Bound constraints: :math:`x \\in [-10, 10]`
.. figure:: figures/Problem03.png
:alt: Univariate Problem03 function
:align: center
**Univariate Problem03 function**
*Global optimum*: :math:`f(x)=-12.03124` for :math:`x = -6.7745761`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(-10, 10)]
self.global_optimum = -6.7745761
self.fglob = -12.03124
def fun(self, x, *args):
self.nfev += 1
x = x[0]
y = 0.0
for k in range(1, 6):
y += k * sin((k + 1) * x + k)
return -y
class Problem04(Benchmark):
"""
Univariate Problem04 objective function.
This class defines the Univariate Problem04 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem04}}(x) = - \\left(16x^2 - 24x + 5 \\right) e^{-x}
Bound constraints: :math:`x \\in [1.9, 3.9]`
.. figure:: figures/Problem04.png
:alt: Univariate Problem04 function
:align: center
**Univariate Problem04 function**
*Global optimum*: :math:`f(x)=-3.85045` for :math:`x = 2.868034`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(1.9, 3.9)]
self.global_optimum = 2.868034
self.fglob = -3.85045
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return -(16 * x ** 2 - 24 * x + 5) * exp(-x)
class Problem05(Benchmark):
"""
Univariate Problem05 objective function.
This class defines the Univariate Problem05 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem05}}(x) = - \\left(1.4 - 3x \\right) \\sin(18x)
Bound constraints: :math:`x \\in [0, 1.2]`
.. figure:: figures/Problem05.png
:alt: Univariate Problem05 function
:align: center
**Univariate Problem05 function**
*Global optimum*: :math:`f(x)=-1.48907` for :math:`x = 0.96609`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(0.0, 1.2)]
self.global_optimum = 0.96609
self.fglob = -1.48907
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return -(1.4 - 3 * x) * sin(18.0 * x)
class Problem06(Benchmark):
"""
Univariate Problem06 objective function.
This class defines the Univariate Problem06 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem06}}(x) = - \\left[x + \\sin(x) \\right] e^{-x^2}
Bound constraints: :math:`x \\in [-10, 10]`
.. figure:: figures/Problem06.png
:alt: Univariate Problem06 function
:align: center
**Univariate Problem06 function**
*Global optimum*: :math:`f(x)=-0.824239` for :math:`x = 0.67956`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(-10.0, 10.0)]
self.global_optimum = 0.67956
self.fglob = -0.824239
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return -(x + sin(x)) * exp(-x ** 2.0)
class Problem07(Benchmark):
"""
Univariate Problem07 objective function.
This class defines the Univariate Problem07 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem07}}(x) = \\sin(x) + \\sin \\left(\\frac{10}{3}x \\right) + \\log(x) - 0.84x + 3
Bound constraints: :math:`x \\in [2.7, 7.5]`
.. figure:: figures/Problem07.png
:alt: Univariate Problem07 function
:align: center
**Univariate Problem07 function**
*Global optimum*: :math:`f(x)=-1.6013` for :math:`x = 5.19978`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(2.7, 7.5)]
self.global_optimum = 5.19978
self.fglob = -1.6013
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return sin(x) + sin(10.0 / 3.0 * x) + log(x) - 0.84 * x + 3
class Problem08(Benchmark):
"""
Univariate Problem08 objective function.
This class defines the Univariate Problem08 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem08}}(x) = - \\sum_{k=1}^6 k \\cos[(k+1)x+k]
Bound constraints: :math:`x \\in [-10, 10]`
.. figure:: figures/Problem08.png
:alt: Univariate Problem08 function
:align: center
**Univariate Problem08 function**
*Global optimum*: :math:`f(x)=-14.508` for :math:`x = -7.083506`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(-10, 10)]
self.global_optimum = -7.083506
self.fglob = -14.508
def fun(self, x, *args):
self.nfev += 1
x = x[0]
y = 0.0
for k in range(1, 6):
y += k * cos((k + 1) * x + k)
return -y
class Problem09(Benchmark):
"""
Univariate Problem09 objective function.
This class defines the Univariate Problem09 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem09}}(x) = \\sin(x) + \\sin \\left(\\frac{2}{3} x \\right)
Bound constraints: :math:`x \\in [3.1, 20.4]`
.. figure:: figures/Problem09.png
:alt: Univariate Problem09 function
:align: center
**Univariate Problem09 function**
*Global optimum*: :math:`f(x)=-1.90596` for :math:`x = 17.039`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(3.1, 20.4)]
self.global_optimum = 17.039
self.fglob = -1.90596
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return sin(x) + sin(2.0 / 3.0 * x)
class Problem10(Benchmark):
"""
Univariate Problem10 objective function.
This class defines the Univariate Problem10 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem10}}(x) = -x\\sin(x)
Bound constraints: :math:`x \\in [0, 10]`
.. figure:: figures/Problem10.png
:alt: Univariate Problem10 function
:align: center
**Univariate Problem10 function**
*Global optimum*: :math:`f(x)=-7.916727` for :math:`x = 7.9787`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(0, 10)]
self.global_optimum = 7.9787
self.fglob = -7.916727
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return -x * sin(x)
class Problem11(Benchmark):
"""
Univariate Problem11 objective function.
This class defines the Univariate Problem11 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem11}}(x) = 2\\cos(x) + \\cos(2x)
Bound constraints: :math:`x \\in [-\\pi/2, 2\\pi]`
.. figure:: figures/Problem11.png
:alt: Univariate Problem11 function
:align: center
**Univariate Problem11 function**
*Global optimum*: :math:`f(x)=-1.5` for :math:`x = 2.09439`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(-pi / 2, 2 * pi)]
self.global_optimum = 2.09439
self.fglob = -1.5
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return 2 * cos(x) + cos(2 * x)
class Problem12(Benchmark):
"""
Univariate Problem12 objective function.
This class defines the Univariate Problem12 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem12}}(x) = \\sin^3(x) + \\cos^3(x)
Bound constraints: :math:`x \\in [0, 2\\pi]`
.. figure:: figures/Problem12.png
:alt: Univariate Problem12 function
:align: center
**Univariate Problem12 function**
*Global optimum*: :math:`f(x)=-1` for :math:`x = \\pi`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(0, 2 * pi)]
self.global_optimum = pi
self.fglob = -1
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return (sin(x)) ** 3.0 + (cos(x)) ** 3.0
class Problem13(Benchmark):
"""
Univariate Problem13 objective function.
This class defines the Univariate Problem13 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem13}}(x) = -x^{2/3} - (1 - x^2)^{1/3}
Bound constraints: :math:`x \\in [0.001, 0.99]`
.. figure:: figures/Problem13.png
:alt: Univariate Problem13 function
:align: center
**Univariate Problem13 function**
*Global optimum*: :math:`f(x)=-1.5874` for :math:`x = 1/\\sqrt(2)`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(0.001, 0.99)]
self.global_optimum = 1.0 / sqrt(2)
self.fglob = -1.5874
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return -x ** (2.0 / 3.0) - (1.0 - x ** 2) ** (1.0 / 3.0)
class Problem14(Benchmark):
"""
Univariate Problem14 objective function.
This class defines the Univariate Problem14 global optimization problem. This
is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem14}}(x) = -e^{-x} \\sin(2\\pi x)
Bound constraints: :math:`x \\in [0, 4]`
.. figure:: figures/Problem14.png
:alt: Univariate Problem14 function
:align: center
**Univariate Problem14 function**
*Global optimum*: :math:`f(x)=-0.788685` for :math:`x = 0.224885`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(0.0, 4.0)]
self.global_optimum = 0.224885
self.fglob = -0.788685
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return -exp(-x) * sin(2.0 * pi * x)
class Problem15(Benchmark):
"""
Univariate Problem15 objective function.
This class defines the Univariate Problem15 global optimization problem.
This is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem15}}(x) = \\frac{x^{2} - 5 x + 6}{x^{2} + 1}
Bound constraints: :math:`x \\in [-5, 5]`
.. figure:: figures/Problem15.png
:alt: Univariate Problem15 function
:align: center
**Univariate Problem15 function**
*Global optimum*: :math:`f(x)=-0.03553` for :math:`x = 2.41422`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(-5.0, 5.0)]
self.global_optimum = 2.41422
self.fglob = -0.03553
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return -(-x ** 2.0 + 5 * x - 6) / (x ** 2 + 1)
class Problem18(Benchmark):
"""
Univariate Problem18 objective function.
This class defines the Univariate Problem18 global optimization problem.
This is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem18}}(x) = \\begin{cases}(x-2)^2 & \\textrm{if} \\hspace{5pt} x \\leq 3 \\\\
2\\log(x-2)+1&\\textrm{otherwise}\\end{cases}
Bound constraints: :math:`x \\in [0, 6]`
.. figure:: figures/Problem18.png
:alt: Univariate Problem18 function
:align: center
**Univariate Problem18 function**
*Global optimum*: :math:`f(x)=0` for :math:`x = 2`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(0.0, 6.0)]
self.global_optimum = 2
self.fglob = 0
def fun(self, x, *args):
self.nfev += 1
x = x[0]
if x <= 3:
return (x - 2.0) ** 2.0
return 2 * log(x - 2.0) + 1
class Problem20(Benchmark):
"""
Univariate Problem20 objective function.
This class defines the Univariate Problem20 global optimization problem.
This is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem20}}(x) = -[x-\\sin(x)]e^{-x^2}
Bound constraints: :math:`x \\in [-10, 10]`
.. figure:: figures/Problem20.png
:alt: Univariate Problem20 function
:align: center
**Univariate Problem20 function**
*Global optimum*: :math:`f(x)=-0.0634905` for :math:`x = 1.195137`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(-10, 10)]
self.global_optimum = 1.195137
self.fglob = -0.0634905
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return -(x - sin(x)) * exp(-x ** 2.0)
class Problem21(Benchmark):
"""
Univariate Problem21 objective function.
This class defines the Univariate Problem21 global optimization problem.
This is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem21}}(x) = x \\sin(x) + x \\cos(2x)
Bound constraints: :math:`x \\in [0, 10]`
.. figure:: figures/Problem21.png
:alt: Univariate Problem21 function
:align: center
**Univariate Problem21 function**
*Global optimum*: :math:`f(x)=-9.50835` for :math:`x = 4.79507`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(0, 10)]
self.global_optimum = 4.79507
self.fglob = -9.50835
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return x * sin(x) + x * cos(2.0 * x)
class Problem22(Benchmark):
"""
Univariate Problem22 objective function.
This class defines the Univariate Problem22 global optimization problem.
This is a multimodal minimization problem defined as follows:
.. math::
f_{\\text{Problem22}}(x) = e^{-3x} - \\sin^3(x)
Bound constraints: :math:`x \\in [0, 20]`
.. figure:: figures/Problem22.png
:alt: Univariate Problem22 function
:align: center
**Univariate Problem22 function**
*Global optimum*: :math:`f(x)=e^{-27\\pi/2} - 1` for :math:`x = 9\\pi/2`
"""
def __init__(self, dimensions=1):
Benchmark.__init__(self, dimensions)
self._bounds = [(0, 20)]
self.global_optimum = 9.0 * pi / 2.0
self.fglob = exp(-27.0 * pi / 2.0) - 1.0
def fun(self, x, *args):
self.nfev += 1
x = x[0]
return exp(-3.0 * x) - (sin(x)) ** 3.0