polarcbo.objectives.Rastrigin

class polarcbo.objectives.Rastrigin(b=0.0, c=0.0)

Bases: objective

Rastrigin’s function

Rastrigin’s function is a multimodal function with a global minima at \((0,0)\). The function is originally defined on \(\mathbb{R}^2\) as

\[f(x,y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2.\]

See Rastrigin’s function. For our case we employ a shifted version on \(\mathbb{R}^d\), where the global minimum is at \((b)\) and we additonally employ a offset \(c\),

\[\tilde{f}(x,y) = \frac{1}{n} \sum_{i=1}^n (x_i - b)^2 - 10 \cos(2 \pi (x_i - b)) + 10 + c.\]

Parameters:

• b (float, optional) – The first parameter of the function. The default is 0.0.

• c (float, optional) – The second parameter of the function. The default is 0.0.

Examples

>>> import numpy as np
>>> from polarcbo.objectives import Rastrigin
>>> x = np.array([[1,2], [3,4], [5,6]])
>>> f = Rastrigin()
>>> f(x)
array([  68.,  148., 1556.])

__call__(x)

Call method for classes that inherit from objective

Parameters:

x (array_like, shape (J, d)) – For a system of \(J\) particles, the i-th row of this array x[i,:] represents the position of the i-th particle.

Returns:

y – The value of the objective function at the positions x.

Return type:

array_like, shape (J,)