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Schwefel Function

Updated: Jul 18, 2021






Mathematical Definition




Input Domain


It can define into any input domain but itโ€™s usually evaluates on the hypercube xiโˆˆ[-500,500] for ๐‘–=1,โ€ฆ,๐‘‘. i=1,โ€ฆ,d.


Global Minima


It has global minimum value ๐‘“(๐‘ฅโˆ—)=0, at xโˆ— = (420.9687,โ€ฆ,420.9687)


Description and Features


The schwefel function is continuous function.

The schwefel function is complex function.

The function can be defined on n - dimensional space.

The schwefel function is multimodal.



Python Implementation


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import matplotlib.pyplot as plt
import numpy as np
from numpy import *
from numpy import meshgrid
from mpl_toolkits.mplot3d import Axes3D

def schwefel( x1,x2):  
    return 418.9829*2 - x1 * sin( sqrt( abs( x1 )))-x2*sin(sqrt(abs(x2)))

x1=linspace(-500,500)
x2=linspace(-500,500)
r_min,r_max=-500,500

x1,x2=np.meshgrid(x1,x2)
results=schwefel(x1,x2)
figure=plt.figure()
axis=figure.gca(projection='3d')
axis.plot_surface(x1,x2,results,color='green')
axis.plot_wireframe(x1,x2,results,color='rgb')
axis.set_xlabel('X')
axis.set_ylabel('Y')
axis.set_zlabel('Z')
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

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