Dixon-Szegö's Function

I. Mathematical Expression:

$$f(x)=4x^2-4x^3+x^4$$

where:

\(\bullet\) \(-5 \leq x \leq 5\)

\(\bullet\) It has two global minimum: \(f_{min}(x^*)=0\)

\(\bullet\) \(x^*=\{0, 2\}\)

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:

% Dixon-Szego's Function
% Range of initial points: -5 <= x <= 5
% Two global minimum: x={0,2}
% f(x)=0
% Coded by: Ali R. Alroomi | Last Update: 10 August 2015 | www.al-roomi.org
 
clear
clc
warning off
 
xmin=-5;
xmax=5;
R=100000; % steps resolution
x=xmin:(xmax-xmin)/R:xmax;
 
for i=1:length(x)
    f(i)=4*x(i).^2-4*x(i).^3+x(i).^4;
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 et al., Handbook of Test Problems in Local and Global Optimization (Noncovex Optimization and Its Applications, vol. 33). Dordrecht, The Netherlands: Kluwer Academic Publishers, 1999.
[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