Zilinskas' Function No.02

I. Mathematical Expression:

$$f(x)=2(x-0.75)^2+\sin\left(\displaystyle 8\pi x -\frac{\pi}{2} \right)-0.125$$

where:

\(\bullet\) \(-2 \leq x \leq 2\) \(\rightarrow\) sometimes it is taken as \(0 \leq x \leq 1\)

\(\bullet\) \(f_{min}(x^*)=-1.125\)

\(\bullet\) \(x^* = 0.75\)

II. Citation Policy:

If you publish material based on databases obtained from this repository, then, in your acknowledgments, please note the assistance you received by using this repository. This will help others to obtain the same data sets and replicate your experiments. We suggest the following pseudo-APA reference format for referring to this repository:

Ali R. Al-Roomi (2015). Unconstrained Single-Objective Benchmark Functions Repository [https://www.al-roomi.org/benchmarks/unconstrained]. Halifax, Nova Scotia, Canada: Dalhousie University, Electrical and Computer Engineering.

Here is a BiBTeX citation as well:

@MISC{Al-Roomi2015,
author = {Ali R. Al-Roomi},
title = {{Unconstrained Single-Objective Benchmark Functions Repository}},
year = {2015},
address = {Halifax, Nova Scotia, Canada},
institution = {Dalhousie University, Electrical and Computer Engineering},
url = {https://www.al-roomi.org/benchmarks/unconstrained}
}

III. 2D-Plot:

IV. MATLAB M-File:

% Zilinskas' Function # 2 
% Range of initial points: -2 <= x <= 2 or 0 <= x <= 1
% Global minima: x=0.75
% f(x)=-1.125
% Coded by: Ali R. Alroomi | Last Update: 13 Jan. 2015 | www.al-roomi.org
 
clear
clc
warning off
 
xmin=-2;
xmax=2;
R=100000; % steps resolution
x=xmin:(xmax-xmin)/R:xmax;
    
for i=1:length(x)
    f(i)=2*(x(i)-0.75).^2+sin(8*pi*x(i)-(pi/2))-0.125;
end
 
plot(x,f,'r','LineWidth',2);grid;set(gca,'FontSize',12);
xlabel('x','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('f(x)','FontName','Times','FontSize',20,'FontAngle','italic');

Click here to download m-file

V. References:

[1] E. Kiseleva and T. Stepanchuk, “On the Efficiency of a Global Non-Differentiable Optimization Algorithm Based on the Method of Optimal Set Partitioning," Journal of Global Optimization, vol. 25, no. 2, pp. 209-235, Feb. 2003.
[2] Ali R. Alroomi, "The Farm of Unconstrained Benchmark Functions," University of Bahrain, Electrical and Electronics Department, Bahrain, Oct. 2013. [Online]. Available: http://www.al-roomi.org/cv/publications