Pseudoethane Function

I. Mathematical Expression:

$$f(t)=f1-f2+f3-f4+f5-f6$$

where:

\(\bullet\) \(f1(t)=\frac{\displaystyle 588600}{\displaystyle \left\{3r^2_0-4r^2_0\cos(\theta)-2\left[\sin^2(\theta)\cos(t-\frac{2 \pi}{3})-\cos^2(\theta)\right]r^2_0\right\}^6}\)

\(\bullet\) \(f2(t)=\frac{\displaystyle 1079.1}{\displaystyle \left\{3r^2_0-4r^2_0\cos(\theta)-2\left[\sin^2(\theta)\cos(t-\frac{2 \pi}{3})-\cos^2(\theta)\right]r^2_0\right\}^3}\)

\(\bullet\) \(f3(t)=\frac{\displaystyle 600800}{\displaystyle \left\{3r^2_0-4r^2_0\cos(\theta)-2\left[\sin^2(\theta)\cos(t)-\cos^2(\theta)\right]r^2_0\right\}^6}\)

\(\bullet\) \(f4(t)=\frac{\displaystyle 1071.5}{\displaystyle \left\{3r^2_0-4r^2_0\cos(\theta)-2\left[\sin^2(\theta)\cos(t)-\cos^2(\theta)\right]r^2_0\right\}^3}\)

\(\bullet\) \(f5(t)=\frac{\displaystyle 481300}{\displaystyle \left\{3r^2_0-4r^2_0\cos(\theta)-2\left[\sin^2(\theta)\cos(t+\frac{2 \pi}{3})-\cos^2(\theta)\right]r^2_0\right\}^6}\)

\(\bullet\) \(f6(t)=\frac{\displaystyle 1064.6}{\displaystyle \left\{3r^2_0-4r^2_0\cos(\theta)-2\left[\sin^2(\theta)\cos(t+\frac{2 \pi}{3})-\cos^2(\theta)\right]r^2_0\right\}^3}\)

\(\bullet\) \(r_0=1.54\) and \(\theta=109.5^0 \times\frac{\displaystyle \pi}{\displaystyle 180^0}\)

\(\bullet\) \(0 \leq t \leq 2\pi\)

\(\bullet\) \(f_{min}(t^*)=-1.071114593111043\)

\(\bullet\) \(t^*=3.201787176863421\)

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:

% Pseudoethane Function
% Range of initial points: 0 <= x <= 2*pi
% Global minima: x=3.201787176863421
% f(x)=-1.071114593111043
% Coded by: Ali R. Alroomi | Last Update: 05 March 2015 | www.al-roomi.org
 
clear
clc
warning off
 
r0=1.54;
theta=109.5*(pi/180);
 
xmin=0;
xmax=2*pi;
R=100000; % steps resolution
x=xmin:(xmax-xmin)/R:xmax;
 
for i=1:length(x)
    f1=588600/((3*r0^2-4*r0^2*cos(theta)-2*((sin(theta)^2)*cos(x(i)-(2*pi/3))-cos(theta)^2)*r0^2)^6);
    f2=1079.1/((3*r0^2-4*r0^2*cos(theta)-2*((sin(theta)^2)*cos(x(i)-(2*pi/3))-cos(theta)^2)*r0^2)^3);
    f3=600800/((3*r0^2-4*r0^2*cos(theta)-2*((sin(theta)^2)*cos(x(i))-cos(theta)^2)*r0^2)^6);
    f4=1071.5/((3*r0^2-4*r0^2*cos(theta)-2*((sin(theta)^2)*cos(x(i))-cos(theta)^2)*r0^2)^3);
    f5=481300/((3*r0^2-4*r0^2*cos(theta)-2*((sin(theta)^2)*cos(x(i)+(2*pi/3))-cos(theta)^2)*r0^2)^6);
    f6=1064.6/((3*r0^2-4*r0^2*cos(theta)-2*((sin(theta)^2)*cos(x(i)+(2*pi/3))-cos(theta)^2)*r0^2)^3);
    f(i)=f1-f2+f3-f4+f5-f6;
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] C. A. Floudas, P. M. Pardalos, C. Adjiman, W. R. Esposito, Z. H. Gümüs, S. T. Harding, J. L. Klepeis, C. A. Meyer, and C. A. Schweiger, Handbook of Test Problems in Local and Global Optimization, ser. Nonconvex Optimization and Its Applications (vol.33). Princeton, New Jersey: Springer, 1999.
[2] Neculai Andrei, Nonlinear Optimization Applications Using the GAMS Technology. : Springer US, 2013.
[3] 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