Mathematical Definition

Input Domain

The Keane Function is defined on input range x [0,10] and y [0,10].

Global Minima

The Keane Function has two global minimum f(x*) = 0.673667521146855 at

x* = (1.393249070031784, 0)

x* = (0, 1.393249070031784)

Description and Features

The Keane Function is defined on 2-dimensional space. This function is considered as standard benchmark for non-linear constrained optimization.

The Keane Function is

Non-separable

Continuous

Differentiable

Non-convex

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 sqrt
def f(x1,x2):
a=-sin(x1-x2)**2*sin(x1+x2)**2
b=sqrt(x1*x1+x2*x2)
c=a/b
return c
x1 = np.linspace(10,-10)
x2 = np.linspace(10,-10)
x1,x2=np.meshgrid(x1,x2)
F=f(x1,x2)
fig =plt.figure(figsize=(9,9))
ax=fig.gca(projection='3d')
ax.contour3D(x1,x2,F,450)
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.set_title('Keane Function')
ax.view_init(21,45)
#plt.contour(x1,x2,F,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.

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