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
% Please forward any comments or bug reports in chat
Copyright 2021. INDUSMIC PRIVATE LIMITED.THERE IS NO WARRANTY, EXPRESS OR IMPLIED. WE DO NOT ASSUME ANY LIABILITY FOR THE USE OF THIS PROGRAM. If software is modified to produce derivative works, such modified software should be clearly marked. Additionally, user can redistribute it and/or modify it under the terms of the GNU General Public License. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY. See the GNU General Public License for more details.
% for any support connect with us on help.indusmic@gmail.com
% Author: Ayushi Manish Shukla
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
f(1,1)
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.plot_surface(X1,X2,f(X1,X2),cmap=cm.coolwarm,
alpha=0.9)
ax.plot_wireframe(X1,X2,f(X1,X2),
alpha=1,rcount=15,ccount=15)
ax.view_init(elev=E, azim=A)
ax.set_xlabel ('X')
ax.set_xlabel ('Y')
ax.set_zlabel ('f(X, Y)')
plotter(45,45)
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.