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DECKKERS-AARTS FUNCTION





Mathematical Definition



Input Domain


The function is usually evaluated for the range x ∈ [-20,20] and y ∈ [-20,20].



Global Minima


The global minima f(x*) =−24771.09375 are located at x*= (0, ±15)


Description and Features


  • The function is continuous.

  • The function is not convex.

  • The function is defined on 2-dimensional space.

  • The function is multimodal.

  • The function is differentiable.

  • The function is non-separable.


Python Implementation


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%matplotlib inline
import math
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

%matplotlib notebook
plt.rcParams['figure.figsize'] = (6,4)
plt.rcParams['figure.dpi']=150
fig=plt.figure()
ax=fig.add_subplot(111,projection='3d')

def z_function(x,y):
 return (10**5)*(x**2)+(y**2)-((x**2)+(y**2))**2+(10**(-5))*((x**2)+(y**2))**4
x= np.linspace(-20,20,100)
y= np.linspace(-20,20,100)
X,Y= np.meshgrid(x,y)
Z= z_function(X,Y)
ax.set_xlabel("(x-axis)")
ax.set_ylabel("(y-axis)")
ax.set_zlabel("(z-axis)")
ax.plot_surface(X,Y,Z,cmap='viridis')

def plotter(E,A):
 ax.view_init(elev=E,azim=A)
 ax.set_title("Deckkers-Aarts Function")
 plt.show()
 plt.contour(X,Y,Z)
#plotter(90,45)

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

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