Discontinuous Function

I. Mathematical Expression:

$$f(x)=\begin{cases}
\frac{\displaystyle x}{\displaystyle 3}\sin(x)\cos(x) & \text{ if } x\in [0,6) \\
\frac{\displaystyle x}{\displaystyle 4}\sin(0.9x) & \text{ otherwise }
\end{cases}$$

where:

\(\bullet\) \(0 \leq x \leq 10\)

\(\bullet\) \(f_{min}(x^*)=-1.159146731333981\)

\(\bullet\) \(x^*=6\)

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:

% Discontinuous Function
% Range of initial points: 0 <= x <= 10
% Global minima: x=6
% f(x)=-1.159146731333981
% Coded by: Ali R. Alroomi | Last Update: 02 March 2015 | www.al-roomi.org
 
clear
clc
warning off
 
xmin=0;
xmax=10;
R=100000; % steps resolution
x=xmin:(xmax-xmin)/R:xmax;
 
for i=1:length(x)
    
    if x(i)<6
        x1(i)=x(i);
        f1(i)=(x(i)/3)*sin(x(i))*cos(x(i));
    else
        x2(i-length(x1))=x(i);
        f2(i-length(x1))=(x(i)/4)*sin(0.9*x(i));
    end
    
end
 
hold;grid;set(gca,'FontSize',12);
plot(x1,f1,'r','LineWidth',2);
plot(x1(end),f1(end),'.r', 'MarkerSize', 30);
plot(x2,f2,'r','LineWidth',2);
plot(x2(1),f2(1),'.r', 'MarkerSize', 30);
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] Troy Allen Henderson, "A Learning Approach to Sampling Optimization: Applications in Astrodynamics," Ph.D. Dissertation, Texas A&M University, Texas, 2013, [Accessed May. 2, 2015]. [Online]. Available: http://repository.tamu.edu/bitstream/handle/1969.1/151266/HENDERSON-DISSERTATION-2013.pdf
[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