ZakharovΒΆ
The Zakharov function has no local minima except the global one. It is shown here in its two-dimensional form.
Definition
\begin{align} \begin{split} f(x) &=& \sum\limits_{i=1}^n {x_i^2} + \bigg( \frac{1}{2} \sum\limits_{i=1}^n {ix_i} \bigg)^2 + \bigg( \frac{1}{2} \sum\limits_{i=1}^n {ix_i} \bigg)^4, \\[2mm] && -10 \leq x_i \leq 10 \quad i=1,\ldots,n \end{split} \end{align}
Optimum
\[f(x^*) = 0 \; \text{at} \; x^* = (0,\ldots,0)\]
Fitness Landscape
[1]:
import numpy as np
from pymoo.problems import get_problem
from pymoo.visualization.fitness_landscape import FitnessLandscape
problem = get_problem("zakharov", n_var=2)
FitnessLandscape(problem, angle=(45, 45), _type="surface").show()
[1]:
<pymoo.visualization.fitness_landscape.FitnessLandscape at 0x123db9d60>
[2]:
FitnessLandscape(problem, _type="contour", contour_levels = 200, colorbar=True).show()
[2]:
<pymoo.visualization.fitness_landscape.FitnessLandscape at 0x132673b00>
