QUARTIC FUNCTION

Mathematical Definition

Input Domain

The function can be defined on any input domain but it is usually evaluated on xi ∈[−1.28,1.28] for i=1,…,n. Here, n = 2.

Global Minima

The function has one global minimum f(x*)=0+random noise at x*=(0,…,0).

Description and Features

The function is continuous, not convex, defined on n-dimensional space, multimodal, differentiable, separable.

Python Implementation

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import numpy as np
import matplotlib.pyplot as plt
from numpy import random
fig = plt.figure()
ax = fig.gca(projection='3d')
x = np.arange(-1.28, 1.28,0.1)
y = np.arange(-1.28, 1.28, 0.1)
x, y = np.meshgrid(x, y)
z = random.randint(0,1)
sum =0
for i in range(3):
    sum = sum + (i * (x**4 + y **4))
result = sum + z
surface = ax.plot_surface(x, y, result, cmap='jet')
plt.show()
plt.contour(x,y,result)
plt.show()
plt.scatter(x, y, result)
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.

#optimization #benchmarkfunction