Mathematical Definition
Input Domain
The Periodic Function is defined on input range xi [-10,10] for i=1…n
Global Minima
The Periodic Function has on global minimum f(x*)=0.9 at x*=(0…0).
Description and Features
The Periodic Function is a function that repeats its values at regular intervals. This function is defined on n-dimensional space and used to describe oscillations, waves and other phenomena that exhibit periodicity.
The Periodic Function is a
Non-separable
Continuous
Non-convex
Differentiable
Multi-modal
Python Implementation
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% Author: RIYA SHIVHARE
import matplotlib.pyplot as plt
import numpy as np
from numpy import sin
from numpy import exp
from mpl_toolkits import mplot3d
def f(x1,x2):
return 1+ sin(x1)**2 + sin(x2)**2-0.1*exp(-x1*x1-x2*x2)
x1=np.linspace(-2,2)
x2=np.linspace(-2,2)
x1,x2=np.meshgrid(x1,x2)
results=f(x1,x2)
fig =plt.figure(figsize=(9,9))
axis=fig.gca(projection='3d')
axis.contour3D(x1,x2,results,450)
axis.set_xlabel('X')
axis.set_ylabel('Y')
axis.set_zlabel('Z')
axis.set_title('Periodic Function')
axis.view_init(21,45)
#plt.contour(x1,x2,results,15)
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.