Mathematical Definition
Input Domain
The function is usually evaluated on the hypercube xi ∈ [-5, 10], for all i = 1, …, d.
Global Minima
f(x*)=0, at x* =(0,…,0).
Description
The function is continuous.
The function is convex.
The function can be defined on n-dimensional space.
The function is unimodal.
The function is differentiable.
The function is non-separable.
Python Implementation
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# Author: Dhanishtha Sharma
import math
import numpy as np
from numpy import *
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 f(x1,x2):
a= 0.5*x1+x2
b= x1*x1 +x2*x2 +pow(a,2) +pow(a,4)
return b
x1= np.linspace(-5,10)
x2= np.linspace(-5,10)
X1,X2= meshgrid(x1,x2)
ax.plot_surface(X1,X2,f(X1,X2), cmap='jet')
ax.set_xlabel('x1')
ax.set_ylabel('x2')
ax.set_zlabel('f(x1,x2)')
ax.view_init(10,10)
plt.show()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.