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Updated: Aug 5, 2021

Mathematical Definition

Input Domain

The function can be defined on any input domain but it is usually evaluated on xi ∈ [−5, 10], xi ∈ [−5, 10] for i=1,…, n.

Global Minima

The function has one global minimum f(x∗)=0 at x∗=(1,…,1).

Description and Features

  • The function is continuous.

  • The function is convex.

  • The function is defined on n-dimensional space.

  • The function is multimodal.

  • The function is differentiable.

  • The function is non-separable.

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 100*(x2-x1**2)**2+(x1-1)**2
x1 = np.linspace(-5, 10)
x2 = np.linspace(-5, 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='jet', alpha=0.8)
  ax.plot_wireframe(X1, X2, f(X1, X2), rcount=15, ccount=15)
  ax.view_init(elev=E, azim=A)
  ax.set_zlabel('f(X, Y)')
  ax.contourf(x1, x2, f(X1, X2))
  print("solution 2")
plotter(45, 45)
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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