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Python Implementation of Shubert N. 4 Function

Updated: Jul 18, 2021





Mathematical Definition



Input Domain


The function can be defined on any input range but it is usually evaluated on xi ∈ [-10,10] for

i = 1,...,n.


Global Minima


The function has one global minimum f(x*) ≈−25.740858.


Characteristics


The Shubert function has several local and global minima.

The function is continuous.

The function is not convex.

The function is defined on n-dimensional space.

The function is multimodal.

The function is differentiable.

The function is separable.



Python Implementation

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% Author: SHIVANGI CHANDRA DUBEY

#For n=2
#shubertn4 accepts the   values of 2 MxM dimension matrices X1, X2
#it returns the   computation of the matrices in an MxM matrix Z
#the function is then   plotted using (X1,X2,Z)
#thus giving us a   contour plot

from mpl_toolkits import mplot3d
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt 
from matplotlib import cm

def shubertn4(x1,x2):
 y=0;
 for j in range(1,6):
 y=y+j*np.cos((j+1)*x1+j)+   j*np.cos((j+1)*x2+j)
 return y

x1=np.linspace(-10,10,20)
x2=np.linspace(-10,10,20)

X1,X2=np.meshgrid(x1,x2)
Z=shubertn4(X1,X2)

def plotFunction(e,a):
 fig=plt.figure(figsize = [12,8])
 ax=plt.axes(projection='3d')
 surf=ax.plot_surface(X1,X2,Z,cmap=cm.coolwarm)
 ax.view_init(elev=e,azim=a)
 ax.set_xlabel('X1')
 ax.set_ylabel('X2')
 ax.set_zlabel('fx')
 ax.set_title('Shubert N. 4 Function')
 fig.colorbar(surf, shrink=0.5, aspect=5)
 plt.show()
 plt.contour(X1, X2, Z)
 plt.show()
 
from ipywidgets import interactive
iplot=interactive(plotFunction,
 e=(-90,90,5),
 a=(-90,90,5))
iplot







References:


[1] Survajonic, Sonja & Bingham, Derek, “Virtual Library of Simulation Experiments”, sfu.ca,

https://www.sfu.ca/~ssurjano/optimization.html


#optimization #benchmarkfunction

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