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Mathematical Definition

Input Domain

The function can be defined on any input domain but it is Usually evaluated on the rectangle x1 ∈ [-3, 3], x2 ∈ [-2, 2].

Global Minima

The function has global minimum f (x*) = -1.0316, at x*= (0.0898,-0.7126) and (-0.0898, 0.7126).

Description and Features

  • The function is continuous.

  • The function is non-scaler.

  • The function is multimodal.

  • The function is differentiable.

  • The function is non-separable.

  • The function is defined on 2-dimensional space.

Python Implementation

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from mpl_toolkits import mplot3d
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import sympy as sy
from matplotlib import cm
def f(x1,x2): 
   return 4*x1**2-2.1*x1**4+(x1**6)/3+x1*x2-4*x2**2+4*x2**4

x1 = np.linspace(-3,3)
x2 = np.linspace(-2,2)
X1,X2 = np.meshgrid(x1, x2)
F = f(x1,x2)
plt.contour(X1, X2, f(X1,X2))
def plotter(E,A):
 fig = plt.figure(figsize=[12,8])
 ax = plt.axes(projection='3d')
 ax.view_init(elev=E, azim=A)
 ax.set_xlabel ('X')
 ax.set_xlabel ('Y')
 ax.set_zlabel ('f(X, Y)')

from ipywidgets import interactive
iplot = interactive(plotter, E = (-90 , 90 ,5),A = (-90 , 90 ,5))


[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.

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