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Brent Function






Mathematical Definition


Input Domain


The function can be defined on any input range but it is usually evaluated on x ∈ [-20,0] and y ∈ [-20,0].


Global Minima


The function has one global minimum at f(x∗) = e-200 located at x∗=(−10,−10).


Characteristics


  • The function is convex.

  • The function is defined on 2-dimensional space.

  • The function is differentiable.

  • The function is non-separable.


Python Implementation


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% Author: SHIVANGI CHANDRA DUBEY

#For   n=2
#brent   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   brent(x,y):
 return   (x+10)**2+(y+10)**2+np.exp(-x**2-y**2);

x=np.linspace(-20,0,20)
y=np.linspace(-20,0,20)

X,Y=np.meshgrid(x,y)
Z=brent(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('Brent 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

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