# encoding=utf8
"""Sphere benchmarks."""
from numpy import abs
from NiaPy.benchmarks.benchmark import Benchmark
__all__ = ['Sphere', 'Sphere2', 'Sphere3']
[docs]class Sphere(Benchmark):
r"""Implementation of Sphere functions.
Date: 2018
Authors: Iztok Fister Jr.
License: MIT
Function: **Sphere function**
:math:`f(\mathbf{x}) = \sum_{i=1}^D x_i^2`
**Input domain:**
The function can be defined on any input domain but it is usually
evaluated on the hypercube :math:`x_i ∈ [0, 10]`, for all :math:`i = 1, 2,..., D`.
**Global minimum:** :math:`f(x^*) = 0`, at :math:`x^* = (0,...,0)`
LaTeX formats:
Inline:
$f(\mathbf{x}) = \sum_{i=1}^D x_i^2$
Equation:
\begin{equation}f(\mathbf{x}) = \sum_{i=1}^D x_i^2 \end{equation}
Domain:
$0 \leq x_i \leq 10$
Reference paper:
Jamil, M., and Yang, X. S. (2013).
A literature survey of benchmark functions for global optimisation problems.
International Journal of Mathematical Modelling and Numerical Optimisation,
4(2), 150-194.
"""
Name = ['Sphere']
[docs] def __init__(self, Lower=-5.12, Upper=5.12):
r"""Initialize of Sphere benchmark.
Args:
Lower (Optional[float]): Lower bound of problem.
Upper (Optional[float]): Upper bound of problem.
See Also:
:func:`NiaPy.benchmarks.Benchmark.__init__`
"""
Benchmark.__init__(self, Lower, Upper)
[docs] @staticmethod
def latex_code():
r"""Return the latex code of the problem.
Returns:
str: Latex code
"""
return r'''$f(\mathbf{x}) = \sum_{i=1}^D x_i^2$'''
[docs] def function(self):
r"""Return benchmark evaluation function.
Returns:
Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]: Fitness function
"""
def evaluate(D, sol):
r"""Fitness function.
Args:
D (int): Dimensionality of the problem
sol (Union[int, float, List[int, float], numpy.ndarray]): Solution to check.
Returns:
float: Fitness value for the solution.
"""
val = 0.0
for i in range(D): val += sol[i] ** 2
return val
return evaluate
[docs]class Sphere2(Benchmark):
r"""Implementation of Sphere with different powers function.
Date: 2018
Authors: Klemen Berkovič
License: MIT
Function: **Sun of different powers function**
:math:`f(\textbf{x}) = \sum_{i = 1}^D \lvert x_i \rvert^{i + 1}`
**Input domain:**
The function can be defined on any input domain but it is usually
evaluated on the hypercube :math:`x_i ∈ [-1, 1]`, for all :math:`i = 1, 2,..., D`.
**Global minimum:** :math:`f(x^*) = 0`, at :math:`x^* = (0,...,0)`
LaTeX formats:
Inline:
$f(\textbf{x}) = \sum_{i = 1}^D \lvert x_i \rvert^{i + 1}$
Equation:
\begin{equation} f(\textbf{x}) = \sum_{i = 1}^D \lvert x_i \rvert^{i + 1} \end{equation}
Domain:
$-1 \leq x_i \leq 1$
Reference URL:
https://www.sfu.ca/~ssurjano/sumpow.html
"""
Name = ['Sphere2']
[docs] def __init__(self, Lower=-1., Upper=1.):
r"""Initialize of Sphere2 benchmark.
Args:
Lower (Optional[float]): Lower bound of problem.
Upper (Optional[float]): Upper bound of problem.
See Also:
:func:`NiaPy.benchmarks.Benchmark.__init__`
"""
Benchmark.__init__(self, Lower, Upper)
[docs] @staticmethod
def latex_code():
r"""Return the latex code of the problem.
Returns:
str: Latex code
"""
return r'''$f(\textbf{x}) = \sum_{i = 1}^D \lvert x_i \rvert^{i + 1}$'''
[docs] def function(self):
r"""Return benchmark evaluation function.
Returns:
Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]: Fitness function
"""
def evaluate(D, sol):
r"""Fitness function.
Args:
D (int): Dimensionality of the problem
sol (Union[int, float, List[int, float], numpy.ndarray]): Solution to check.
Returns:
float: Fitness value for the solution.
"""
val = 0.0
for i in range(D): val += abs(sol[i]) ** (i + 2)
return val
return evaluate
[docs]class Sphere3(Benchmark):
r"""Implementation of rotated hyper-ellipsoid function.
Date: 2018
Authors: Klemen Berkovič
License: MIT
Function: **Sun of rotated hyper-elliposid function**
:math:`f(\textbf{x}) = \sum_{i = 1}^D \sum_{j = 1}^i x_j^2`
**Input domain:**
The function can be defined on any input domain but it is usually
evaluated on the hypercube :math:`x_i ∈ [-65.536, 65.536]`, for all :math:`i = 1, 2,..., D`.
**Global minimum:** :math:`f(x^*) = 0`, at :math:`x^* = (0,...,0)`
LaTeX formats:
Inline:
$f(\textbf{x}) = \sum_{i = 1}^D \sum_{j = 1}^i x_j^2$
Equation:
\begin{equation} f(\textbf{x}) = \sum_{i = 1}^D \sum_{j = 1}^i x_j^2 \end{equation}
Domain:
$-65.536 \leq x_i \leq 65.536$
Reference URL:
https://www.sfu.ca/~ssurjano/rothyp.html
"""
Name = ['Sphere3']
[docs] def __init__(self, Lower=-65.536, Upper=65.536):
r"""Initialize of Sphere3 benchmark.
Args:
Lower (Optional[float]): Lower bound of problem.
Upper (Optional[float]): Upper bound of problem.
See Also:
:func:`NiaPy.benchmarks.Benchmark.__init__`
"""
Benchmark.__init__(self, Lower, Upper)
[docs] @staticmethod
def latex_code():
r"""Return the latex code of the problem.
Returns:
str: Latex code
"""
return r'''$f(\textbf{x}) = \sum_{i = 1}^D \sum_{j = 1}^i x_j^2$'''
[docs] def function(self):
r"""Return benchmark evaluation function.
Returns:
Callable[[int, Union[int, float, List[int, float], numpy.ndarray]], float]: Fitness function
"""
def evaluate(D, sol):
r"""Fitness function.
Args:
D (int): Dimensionality of the problem
sol (Union[int, float, List[int, float], numpy.ndarray]): Solution to check.
Returns:
float: Fitness value for the solution.
"""
val = 0.0
for i in range(D):
v = .0
for j in range(i + 1): val += abs(sol[j]) ** 2
val += v
return val
return evaluate
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