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Mathematical Definition

Input Domain

The function is usually evaluated on the hypercube xi ∈ [-10, 10], for all i = 1, …, d.

Global Minima

The function has global minimum

Description and Features

  • The function is continuous.

  • The function is scalable.

  • The function is unimodal.

  • The function is differentiable.

  • The function is non-separable.

  • The function is defined on d-dimensional space.

Python Implementation

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from matplotlib import pyplot as plt
from mpl_toolkits import mplot3d
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
def f(x1, x2): return ((x1-1)**2) + (2*(2*x2**2-x1)**2)
x1 = np.linspace(-1, 10)
x2 = np.linspace(-10, 10)
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.plot_surface(X1, X2, f(X1, X2), cmap='autumn', alpha=0.3)
  ax.view_init(elev=E, azim=A)
  ax.set_zlabel('f(X, Y)')
  ax.contourf(x1, x2, f(X1, X2))
print("solution 2")
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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