Dolan's Function No.01

I. Mathematical Expression:

$$f(x)=x^4-12x^3+15x^2+56x-60$$

where:

\(\bullet\) \(-100 \leq x \leq 100\)

\(\bullet\) \(f_{min}(x^*)=-703.7287810900712\)

\(\bullet\) \(x^*=7.810207524564704\)

\(\bullet\) The given analytical optima in [1] is wrong. Instead, we re-obtained it by using the bracketing technique.

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:

% Dolan's Function # 1
% Range of initial points: -100 <= x <= 100
% Global minima: x=7.810207524564704
% f(x)=-703.7287810900712
% Coded by: Ali R. Alroomi | Last Update: 05 August 2015 | www.al-roomi.org
 
clear
clc
warning off
 
xmin=-100;
xmax=100;
R=100000; % steps resolution
x=xmin:(xmax-xmin)/R:xmax;
 
for i=1:length(x)
    f(i)=x(i)^4-12*x(i)^3+15*x(i)^2+56*x(i)-60;
end
 
plot(x,f,'r','LineWidth',2);grid;set(gca,'FontSize',12);axis([-4 10 -774 2000]);
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] Ariel Dolan, "A General GA Toolkit Implemented in JAVA, for Experimenting with Genetic Algorithms and Handling Optimization Problems," Jun. 1998, [Accessed August 06, 2015]. [Online]. Available: http://www.aridolan.com/ofiles/ga/gaa/gaa.aspx
[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