Benchmarks package

This package contains a set of optimization functions to demonstrate the algorithms ability to deal with multiple modes.

Submodules

star_pso.benchmarks.composite_functions module

star_pso.benchmarks.composite_functions.BASIC_FUNCTIONS: dict = {'f_ackley': CPUDispatcher(<function f_ackley>), 'f_griewank': CPUDispatcher(<function f_griewank>), 'f_rastrigin': CPUDispatcher(<function f_rastrigin>), 'f_sphere': CPUDispatcher(<function f_sphere>), 'f_weierstrass': CPUDispatcher(<function f_weierstrass>)}

Define a dictionary with all the basic functions:

• Ackley

• Sphere

• Griewank

• Rastrigin

• Weierstrass

class star_pso.benchmarks.composite_functions.CompositeFunction(n_dim: int = 2, n_func: int | list = 4, x_min: float = -5.0, x_max: float = 5.0)

Bases: TestFunction

Generates a Composite Function using a weighted average of basic functions, as defined in the ‘BASIC_FUNCTIONS’ dictionary.

static compute_weights(x_pos: ndarray[Any, dtype[_ScalarType_co]], sigma: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

Calculates a set of weights (one for each function).

Parameters:

• x_pos – (ndarray) the position that we are evaluating the functions.

• sigma – (ndarray) are used to control each function’s range. a small value gives a narrow range for the function.

Returns:

a (ndarray) of normalized weights.

func(x_pos: ndarray[Any, dtype[_ScalarType_co]], i_bias: float = 0.0, f_bias: float = 0.0) → float | ndarray[Any, dtype[_ScalarType_co]]

Describes the general framework for the construction of multimodal composition functions with several global optima.

Parameters:

• x_pos – the current position(s) of the function.

• i_bias – function value bias for each basic function.

• f_bias – function value bias for the composition function.

Returns:

the function value(s).

search_for_optima(population: list[Particle], epsilon: float = 0.0001) → tuple[int, int]

Searches the input population for the global optimum values of the specific test function, using default (problem specific) parameters.

Parameters:

• population – the population to search the global optimum.

• epsilon – accuracy level of the global optimal solution.

Returns:

a tuple with the number of global optima found and the total number that exist.

star_pso.benchmarks.composite_functions.f_ackley(x_pos: ndarray[Any, dtype[_ScalarType_co]], alpha: float = 20.0, beta: float = 0.2) → ndarray[Any, dtype[_ScalarType_co]]

Computes the Ackley function at x_pos, with default alpha and beta parameters.

star_pso.benchmarks.composite_functions.f_griewank(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

Computes the Griewank function at x_pos.

star_pso.benchmarks.composite_functions.f_rastrigin(x_pos: ndarray[Any, dtype[_ScalarType_co]], kappa: float = 10.0) → ndarray[Any, dtype[_ScalarType_co]]

Computes the Rastrigin function at x_pos, with default kappa parameter.

star_pso.benchmarks.composite_functions.f_sphere(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

Computes the sphere function at x_pos.

star_pso.benchmarks.composite_functions.f_weierstrass(x_pos: ndarray[Any, dtype[_ScalarType_co]], k_max: int = 9, alpha: float = 0.5, beta: int = 3) → ndarray[Any, dtype[_ScalarType_co]]

Computes the Weierstrass function at x_pos, with default k_max, alpha and beta parameters.

star_pso.benchmarks.equal_maxima module

class star_pso.benchmarks.equal_maxima.EqualMaxima(x_min: float = 0.0, x_max: float = 1.0)

Bases: TestFunction

This function was originally proposed in:

• K. Deb, “Genetic algorithms in multimodal function optimization (master thesis and tcga report no. 89002),” Ph.D. dissertation, Tuscaloosa: University of Alabama, The Clearinghouse for Genetic Algorithms, 1989.

func(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

This is 1D function. There are 5 global optima.

Parameters:

x_pos – the current position(s) of the function.

Returns:

the function value(s).

search_for_optima(population: list[Particle], epsilon: float = 0.0001) → tuple[int, int]

Searches the input population for the global optimum values of the specific test function, using default (problem specific) parameters.

Parameters:

• population – the population to search the global optimum.

• epsilon – accuracy level of the global optimal solution.

Returns:

a tuple with the number of global optima found and the total number that exist.

star_pso.benchmarks.five_uneven_peak_trap module

class star_pso.benchmarks.five_uneven_peak_trap.FiveUnevenPeakTrap(x_min: float = 0.0, x_max: float = 30.0)

Bases: TestFunction

This function was originally proposed in:

• J.-P. Li, M. E. Balazs, G. T. Parks, and P. J. Clarkson, “A species conserving genetic algorithm for multimodal function optimization” Evolutionary Computation, vol. 10, no. 3, pp. 207–234, 2002.

func(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

This is 1D function. There are two global and one local optima.

Parameters:

x_pos – the current position(s) of the function.

Returns:

the function value(s).

search_for_optima(population: list[Particle], epsilon: float = 0.0001) → tuple[int, int]

Searches the input population for the global optimum values of the specific test function, using default (problem specific) parameters.

Parameters:

• population – the population to search the global optimum.

• epsilon – accuracy level of the global optimal solution.

Returns:

a tuple with the number of global optima found and the total number that exist.

star_pso.benchmarks.gaussian_mixture module

class star_pso.benchmarks.gaussian_mixture.GaussianMixture(x_min: float = -15.0, x_max: float = 15.0)

Bases: TestFunction

This function provides a 2D Gaussian mixture model.

The equations are given by the Multivariate Normal Distribution, with four modes (2 global and 2 local):

\[f(x) = \sum_{i=1}^{4} \mathcal{N}(\mu_i, \Sigma_i)\]

with mean vectors:

\[ \begin{align}\begin{aligned}\mu_1 = [-0.0, -1.0]\\\mu_2 = [-4.0, -6.0]\\\mu_3 = [-5.0, +1.0]\\\mu_4 = [5.0, -10.0]\end{aligned}\end{align} \]

and covariances:

\[ \begin{align}\begin{aligned}\Sigma_1 = [ [ 1.0, 0.1 ], [ 0.1, 1.0 ] ]\\\Sigma_2 = [ [ 1.0, 0.1 ], [ 0.1, 1.0 ] ]\\\Sigma_3 = [ [ 1.2, 0.3 ], [ 0.3, 1.2 ] ]\\\Sigma_4 = [ [ 1.2, 0.3 ], [ 0.3, 1.2 ] ]\end{aligned}\end{align} \]

MVN = (<scipy.stats._multivariate.multivariate_normal_frozen object>, <scipy.stats._multivariate.multivariate_normal_frozen object>, <scipy.stats._multivariate.multivariate_normal_frozen object>, <scipy.stats._multivariate.multivariate_normal_frozen object>)

Set up four multivariate normal distributions.

func(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

This is 2D function with is 2 global and 2 local optima.

Parameters:

x_pos – the current position(s) of the function.

Returns:

the function value(s).

search_for_optima(population: list[Particle], epsilon: float = 0.0001) → tuple[int, int]

Searches the input population for the global optimum values of the specific test function, using default (problem specific) parameters.

Parameters:

• population – the population to search the global optimum.

• epsilon – accuracy level of the global optimal solution.

Returns:

a tuple with the number of global optima found and the total number that exist.

star_pso.benchmarks.himmelblau module

class star_pso.benchmarks.himmelblau.Himmelblau(x_min: float = -6.0, x_max: float = 6.0)

Bases: TestFunction

This function was originally proposed in:

• K. Deb, “Genetic algorithms in multimodal function optimization (master thesis and tcga report no. 89002)”, Ph.D. dissertation, Tuscaloosa: University of Alabama, The Clearinghouse for Genetic Algorithms, 1989.

func(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

This is 2D function with is 4 global optima.

Parameters:

x_pos – the current position(s) of the function.

Returns:

the function value(s).

search_for_optima(population: list[Particle], epsilon: float = 0.0001) → tuple[int, int]

Searches the input population for the global optimum values of the specific test function, using default (problem specific) parameters.

Parameters:

• population – the population to search the global optimum.

• epsilon – accuracy level of the global optimal solution.

Returns:

a tuple with the number of global optima found and the total number that exist.

star_pso.benchmarks.rastrigin module

class star_pso.benchmarks.rastrigin.Rastrigin(n_dim: int = 2, x_min: float = 0.0, x_max: float = 1.0)

Bases: TestFunction

This function was originally proposed in:

• A. Saha and K. Deb, “A bi-criterion approach to multimodal optimization: self-adaptive approach”, in Proceedings of the 8th international conference on Simulated evolution and learning, ser. SEAL-10. Berlin, Heidelberg: Springer-Verlag, 2010, pp. 95–104.

func(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

This is a multidimensional function with ‘M’ global optimal values.

Parameters:

x_pos – the current position(s) of the function.

Returns:

the function value(s).

search_for_optima(population: list[Particle], epsilon: float = 0.0001) → tuple[int, int]

Searches the input population for the global optimum values of the specific test function, using default (problem specific) parameters.

Parameters:

• population – the population to search the global optimum.

• epsilon – accuracy level of the global optimal solution.

Returns:

a tuple with the number of global optima found and the total number that exist.

star_pso.benchmarks.shubert module

class star_pso.benchmarks.shubert.Shubert(n_dim: int = 2, x_min: float = -10.0, x_max: float = 10.0)

Bases: TestFunction

This function was originally proposed in:

• Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs. New York: Springer-Verlag, New York, 1996.

func(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

This is a multidimensional function with ‘n_dim * 3^n_dim’ global optimal values.

Parameters:

x_pos – the current position(s) of the function.

Returns:

the function value(s).

search_for_optima(population: list[Particle], epsilon: float = 0.0001) → tuple[int, int]

Searches the input population for the global optimum values of the specific test function, using default (problem specific) parameters.

Parameters:

• population – the population to search the global optimum.

• epsilon – accuracy level of the global optimal solution.

Returns:

a tuple with the number of global optima found and the total number that exist.

star_pso.benchmarks.vincent module

class star_pso.benchmarks.vincent.Vincent(n_dim: int = 2, x_min: float = 0.25, x_max: float = 10.0)

Bases: TestFunction

This function was originally proposed in:

• O. Shir and T. Back, “Niche radius adaptation in the cms-es niching algorithm”, in Parallel Problem-Solving from Nature - PPSN IX, 9th International Conference (LNCS 4193). Reykjavík, Iceland: Springer, 2006, pp. 142 – 151.

func(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

This is a nD function with 6^n_dim global optimal values.

Parameters:

x_pos – the current position(s) of the function.

Returns:

the function value(s).

search_for_optima(population: list[Particle], epsilon: float = 0.0001) → tuple[int, int]

Searches the input population for the global optimum values of the specific test function, using default (problem specific) parameters.

Parameters:

• population – the population to search the global optimum.

• epsilon – accuracy level of the global optimal solution.

Returns:

a tuple with the number of global optima found and the total number that exist.

star_pso.benchmarks.six_hump_camel_back module

class star_pso.benchmarks.six_hump_camel_back.SixHumpCamelBack(x_min: list | ndarray[Any, dtype[_ScalarType_co]] | None = None, x_max: list | ndarray[Any, dtype[_ScalarType_co]] | None = None)

Bases: TestFunction

This function was originally proposed in:

• Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs. New York: Springer-Verlag, New York, 1996.

func(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → float | ndarray[Any, dtype[_ScalarType_co]]

This is a 2D function with 2 global and 2 local optima.

Parameters:

x_pos – the current position(s) of the function.

Returns:

the function value(s).

search_for_optima(population: list[Particle], epsilon: float = 0.0001) → tuple[int, int]

Searches the input population for the global optimum values of the specific test function, using default (problem specific) parameters.

Parameters:

• population – the population to search the global optimum.

• epsilon – accuracy level of the global optimal solution.

Returns:

a tuple with the number of global optima found and the total number that exist.

star_pso.benchmarks.test_function module

class star_pso.benchmarks.test_function.TestFunction(name: str, n_dim: int, x_min: float | ndarray[Any, dtype[_ScalarType_co]], x_max: float | ndarray[Any, dtype[_ScalarType_co]])

Bases: object

Description:

All benchmark test functions should inherit from this class. Here is provided the interface for all the test problems.

func(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

This method will implement the objective function to be optimized.

Returns:

None.

property n_dim: int

Accessor (getter) of the test function dimensions.

Returns:

(int) number of dimensions.

property name: str

Accessor (getter) of the test function name.

Returns:

string name of the test function.

rng: Generator = Generator(PCG64) at 0x7C2F38931C40

Random number generator for the whole class.

sample_random_positions(n_pos: int = 100, method: str = 'random') → ndarray[Any, dtype[_ScalarType_co]]

Generate an initial set of uniformly random sampled positions within the lower / upper bounds of the test problem.

Parameters:

• n_pos – (int) number of random positions to create.

• method – (str) method to use for sampling (“random”, “latin-hc”).

Returns:

a uniformly sampled set of random positions.

search_for_optima(population: list[Particle], epsilon: float = 0.0001) → tuple[int, int]

Searches the input population for the global optimum values of the specific test function, using default (problem specific) parameters.

Parameters:

• population – a list of Particles to search the global optimum.

• epsilon – (float) accuracy level of the global optimal solution.

Returns:

a tuple with the number of global optima found and the total number that exist.

classmethod set_seed(new_seed=None) → None

Sets a new seed for the random number generator.

Parameters:

new_seed – New seed value (default=None).

Returns:

None.

property x_max: float | ndarray[Any, dtype[_ScalarType_co]]

Accessor (getter) of the upper bounds of the test function.

Returns:

numpy array with maximum values.

property x_min: float | ndarray[Any, dtype[_ScalarType_co]]

Accessor (getter) of the lower bounds of the test function.

Returns:

numpy array with minimum values.

star_pso.benchmarks.uneven_decreasing_maxima module

class star_pso.benchmarks.uneven_decreasing_maxima.UnevenDecreasingMaxima(x_min: float = 0.0, x_max: float = 1.0)

Bases: TestFunction

This function was originally proposed in:

• K. Deb, “Genetic algorithms in multimodal function optimization (master thesis and tcga report no. 89002)”, Ph.D. dissertation, Tuscaloosa: University of Alabama, The Clearinghouse for Genetic Algorithms, 1989.

func(x_pos: ndarray[Any, dtype[_ScalarType_co]]) → ndarray[Any, dtype[_ScalarType_co]]

This is 1D function. There is 1 global and 4 local optima.

Parameters:

x_pos – the current position(s) of the function.

Returns:

the function value(s).

search_for_optima(population: list[Particle], epsilon: float = 0.0001) → tuple[int, int]

Searches the input population for the global optimum values of the specific test function, using default (problem specific) parameters.

Parameters:

• population – the population to search the global optimum.

• epsilon – accuracy level of the global optimal solution.

Returns:

a tuple with the number of global optima found and the total number that exist.

Module contents