Xin-She Yang Function

Mathematical Definition

where ∈ is a random number that is drawn uniformly from [0,1]

Input Domain

The function can be defined on any input domain but it is usually evaluated on xi ∈ [−5,5], for i=1,…,n.

Global Minima

The global minima f(x∗) =0 are located at x∗=(0,…,0).

Characteristics

Python Implementation

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 #For n=2
 #xinSheYang 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 xinSheYang(x1,x2):
  return np.random.random(1)*pow(abs(x1),1) +
  np.random.random(1)*pow(abs(x2),2)
 
 x1=np.linspace(-5,5,10)
 x2=np.linspace(-5,5,10)
 
 X1,X2=np.meshgrid(x1,x2)
 Z=xinSheYang(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('Xin-She Yang 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