Problem No.17 (or Lévy's Function No.01)

I. Mathematical Expression:

$$f(x)=-x^6+15x^4-27x^2-250$$

where:

\(\bullet\) \(-4 \leq x \leq 4\)

\(\bullet\) It has two global maximum: \(f_{max}(x^*)=-7\)

\(\bullet\) \(x^*=\{-3,3\}\)

\(\bullet\) Please note that the original reference, which is [1], formulates this function with minimization mode; i.e. \(f_{original}=-f_{used}\).

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:

% Problem # 17 (or Lévy's Function # 1)
% Range of initial points: -4 <= x <= 4
% Two global maxima: x={-3,3}
% f(x)=-7
% Coded by: Ali R. Alroomi | Last Update: 06 July 2015 | www.al-roomi.org
 
clear
clc
warning off
 
xmin=-4;
xmax=4;
R=100000; % steps resolution
x=xmin:(xmax-xmin)/R:xmax;
 
for i=1:length(x)
    f(i)=-x(i)^6+15*x(i)^4-27*x(i)^2-250;
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] A. V. Levy, A. Montalvo, S. Gomez, and A. Calderon, "Topics in Global Optimization," in Lecture Notes in Mathematics (No. 909), A. Dold, and B. Eckmann. Berlin, Heidelberg: Springer Berlin Heidelberg, 1982, pp. 18-33.
[2] Pierre Hansen, Brigitte Jaumard, and Shi-Hui Lu, "Global Optimization of Univariate Lipschitz Functions: II. New Algorithms and Computational Comparison," Mathematical Programming, vol. 55, no. 1-3, pp. 273-292, Apr.1992.
[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