Bartels Conn Function

Mathematical Definition

Input Domain

The function can be defined on any input domain but it is usually evaluated on x ∈ [−500,500], and y ∈ [−500,500].

Global Minima

The global minima f(x∗) =1 is located at x∗ = (0, 0).

Characteristics

The function is not convex.
The function is defined on 2-dimensional space.
The function is non-separable.
The function is non-differentiable.

Python Implementation

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#For n=2
#bartelsConn accepts the values of 2 MxM dimension   matrices X, Y
#it returns the computation of the matrices in an MxM   matrix Z
#the function is then plotted using (X,Y,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 bartelsConn(x,y):
 return   abs(x**2+y**2+x*y)+abs(np.sin(x))+abs(np.cos(y))

x=np.linspace(-500,500,1000)
y=np.linspace(-500,500,1000)

X,Y=np.meshgrid(x,y)
Z=bartelsConn(X,Y)

def plotFunction(e,a): 
 fig=plt.figure(figsize=[12,8])
 ax=plt.axes(projection='3d')
 surf=ax.plot_surface(X,Y,Z,cmap=cm.coolwarm)
 ax.view_init(elev=e,azim=a)
 ax.set_xlabel('X')
 ax.set_ylabel('Y')
 ax.set_zlabel('fx')
 ax.set_title('Bartels Conn Function')
  fig.colorbar(surf,   shrink=0.5, aspect=5)
 plt.show()
 plt.contour(X,Y,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