Python Implementation of DROP WAVE FUNCTION

This function is highly complex with expanding ripples like an object is dropped into liquid

surface . Drop wave function have a global optimum point and have multiple local optimal

regions. So have high chances to mislead search agents.

Mathematical Definition

Input Domain

The function can be defined on any input domain but it is usually evaluated on the square

xi ∈ [-5.12, 5.12], for all i = 1, 2.

Global Minima

f(x0) = -1 , at x0 = (0,0)

Characteristics

The function is continuous.

The function is non-convex.

The function is defined on 2-dimensional space.

The function is unimodal.

The function is differentiable.

The function is non-separable.

The function is non-random.

Python Implementation

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import matplotlib.pyplot as plt
from mpl_toolkits import mplot3d
from numpy import*
 
def f(x1,x2):
    b=0.5*(x1*x1+x2*x2)+2
    a=-(1+cos(12*sqrt(x1*x1+x2*x2)))/b
    return a
x1=linspace(-5.12,5.12,100)
x2=linspace(-5.12,5.12,100)
X1,X2=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='red',alpha=0.7)
    ax.plot_wireframe(X1,X2,f(X1,X2),ccount=2,rcount=2,color='pink', alpha=0.2)
    ax.view_init(elev=E,azim=A)
    ax.set_xlabel('x1')
    ax.set_ylabel('x2')
    ax.set_zlabel('f(x1,x2)')
    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