BIRD FUNCTION

Mathematical Definition

Input Domain

It is evaluated on: x ∈ [-2π, 2π] and y ∈ [-2π,2π]

Global Minima

The function has two global minima at f(x*)= −106.764537 located at x*= (4.70104 ,3.15294) and x*= (−1.58214 ,−3.13024)

Description and Features

• The function is not convex.

• The function is define on 2D space.

• The function is non-separable.

• The function is differentiable.

Python Implementation

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%matplotlib inline
import math
import numpy as np
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 z_function(x,y):
 return np.sin(x)*(np.exp(1-np.cos(y))**2)+np.cos(y)*(np.exp(1-np.sin(x))**2)+(x-y)**2

x= np.linspace(-2*math.pi,2*math.pi,100)
y= np.linspace(-2*math.pi,2*math.pi,100)
X,Y= np.meshgrid(x,y)
Z= z_function(X,Y)
ax.set_xlabel("(x-axis)")
ax.set_ylabel("(y-axis)")
ax.set_zlabel("(z-axis)")
ax.plot_surface(X,Y,Z,cmap='plasma')
ax.contour3D(X,Y,Z)
def plotter(E,A):
 ax.view_init(elev=E,azim=A)
 plt.title("Bird Function")
 plt.show()
 plt.contour(X,Y,Z)
#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.

#optimization #benchmarkfunction