BUKIN FUNCTION N.6

Mathematical Definition

Input Domain

The function is evaluated on the rectangle x1∈[−15,−5] and x2∈[−3,3] but it can be

defined on any input domain.

Global Minima

The function has one global minimum f(x0)=0 at x0=f(−10,1).

Description and Features

Dimension: 2

The Bukin function N.6 has many local minima, all of which lie in a ridge.

Python Implementation

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import matplotlib.pyplot as plt
from mpl_toolkits import mplot3d
import numpy as np

def f(x1,x2):
    ab=np.fabs(x2-0.01*x1*x1)
    a=100*np.sqrt(ab)+0.01*np.fabs(x1+10)
    return a
x1=np.linspace(-15,-5)
x2=np.linspace(-3,3)
X1,X2=np.meshgrid(x1,x2)

def plotter(E,A):
    fig=plt.figure(figsize=[12,8])
    ax=plt.axes(projection='3d')
    ax.plot_surface(X1,X2,f(X1,X2),color='purple',alpha=0.7)
    ax.plot_wireframe(X1,X2,f(X1,X2),ccount=5,rcount=10,color='red', alpha=0.8)
    ax.view_init(elev=E,azim=A)
    ax.set_xlabel('x1')
    ax.set_ylabel('x2')
    ax.set_zlabel('f(x1,x2)')
    ax.set_title('f(x1,x2)=100*sqrt(fabs(x2-0.01*x1*x1)) +0.01*fabs(x1+10)')
    plt.show()

from ipywidgets import interactive
iplot=interactive(plotter,E=(-90,90,5),A=(-90,90,5))
iplot

References:

[1] Jamil, Momin, and Xin-She Yang. "A literature survey of benchmark functions for global optimization problems." International Journal of Mathematical Modelling and Numerical Optimization 4.2 (2013): 150-194.

#optimization #benchmarkfunction