porcupy.functions

Functions

`sphere`(x)	Sphere function for optimization benchmarking.
`rosenbrock`(x)	Rosenbrock function for optimization benchmarking.
`schwefel_2_22`(x)	Schwefel 2.22 function for optimization benchmarking.
`schwefel_1_2`(x)	Schwefel 1.2 function for optimization benchmarking.
`schwefel_2_21`(x)	Schwefel 2.21 function for optimization benchmarking.
`step`(x)	Step function for optimization benchmarking.
`quartic`(x)	Quartic function with noise for optimization benchmarking.
`rastrigin`(x)	Rastrigin function for optimization benchmarking.
`ackley`(x)	Ackley function for optimization benchmarking.
`griewank`(x)	Griewank function for optimization benchmarking.
`schwefel`(x)	Schwefel function for optimization benchmarking.
`michalewicz`(x)	Michalewicz function for optimization benchmarking.
`get_function_by_name`(→ Callable)	Get a benchmark function by name.
`get_function_bounds`(→ Tuple[numpy.ndarray, numpy.ndarray])	Get the recommended bounds for a benchmark function.
`get_function_optimum`(→ Tuple[numpy.ndarray, float])	Get the global optimum for a benchmark function.

Module Contents

porcupy.functions.sphere(x)[source]

Sphere function for optimization benchmarking.

The sphere function is a simple, unimodal function defined as f(x) = sum(x_i^2). Its global minimum is 0 at x = [0, 0, …, 0].

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.rosenbrock(x)[source]

Rosenbrock function for optimization benchmarking.

The Rosenbrock function is a non-convex function defined as f(x) = sum(100 * (x_{i+1} - x_i^2)^2 + (1 - x_i)^2). Its global minimum is 0 at x = [1, 1, …, 1].

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.schwefel_2_22(x)[source]

Schwefel 2.22 function for optimization benchmarking.

This is a unimodal function defined as f(x) = sum(|x_i|) + prod(|x_i|). Its global minimum is 0 at x = [0, 0, …, 0].

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.schwefel_1_2(x)[source]

Schwefel 1.2 function for optimization benchmarking.

This is a unimodal function defined as f(x) = sum(sum(x_j)^2) for j=1 to i, i=1 to n. Its global minimum is 0 at x = [0, 0, …, 0].

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.schwefel_2_21(x)[source]

Schwefel 2.21 function for optimization benchmarking.

This is a unimodal function defined as f(x) = max(|x_i|). Its global minimum is 0 at x = [0, 0, …, 0].

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.step(x)[source]

Step function for optimization benchmarking.

This is a discontinuous function defined as f(x) = sum(floor(x_i + 0.5)^2). Its global minimum is 0 at x in [-0.5, 0.5]^n.

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.quartic(x)[source]

Quartic function with noise for optimization benchmarking.

This is a function defined as f(x) = sum(i * x_i^4) + random[0, 1). Its global minimum is close to 0 at x = [0, 0, …, 0].

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.rastrigin(x)[source]

Rastrigin function for optimization benchmarking.

The Rastrigin function is a multimodal function defined as f(x) = 10 * dim + sum(x_i^2 - 10 * cos(2 * pi * x_i)). Its global minimum is 0 at x = [0, 0, …, 0].

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.ackley(x)[source]

Ackley function for optimization benchmarking.

The Ackley function is a multimodal function defined as f(x) = -20 * exp(-0.2 * sqrt(sum(x_i^2) / n)) - exp(sum(cos(2 * pi * x_i)) / n) + 20 + e. Its global minimum is 0 at x = [0, 0, …, 0].

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.griewank(x)[source]

Griewank function for optimization benchmarking.

The Griewank function is a multimodal function defined as f(x) = 1 + sum(x_i^2 / 4000) - prod(cos(x_i / sqrt(i))). Its global minimum is 0 at x = [0, 0, …, 0].

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.schwefel(x)[source]

Schwefel function for optimization benchmarking.

The Schwefel function is a multimodal function defined as f(x) = 418.9829 * n - sum(x_i * sin(sqrt(|x_i|))). Its global minimum is 0 at x = [420.9687, …, 420.9687].

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.michalewicz(x)[source]

Michalewicz function for optimization benchmarking.

The Michalewicz function is a multimodal function with steep ridges and valleys. It has n! local minima, and the global minimum value depends on the dimension.

Parameters:: x (ndarray) – Input vector (1D array) of shape (dim,).
Returns:: The computed function value.
Return type:: float

porcupy.functions.get_function_by_name(name: str) → Callable[source]

Get a benchmark function by name.

Parameters:: name (str) – Name of the function.
Returns:: The benchmark function.
Return type:: callable
Raises:: ValueError – If the function name is not recognized.

porcupy.functions.get_function_bounds(name: str, dimensions: int) → Tuple[numpy.ndarray, numpy.ndarray][source]

Get the recommended bounds for a benchmark function.

Parameters:

name (str) – Name of the function.
dimensions (int) – Number of dimensions.

Returns:

A tuple (lb, ub) containing the lower and upper bounds.

Return type:

tuple

Raises:

ValueError – If the function name is not recognized.

porcupy.functions.get_function_optimum(name: str, dimensions: int) → Tuple[numpy.ndarray, float][source]

Get the global optimum for a benchmark function.