Problem No.08 (Modified Trigonometric Polynomial or Lévy's Function No.02)

I. Mathematical Expression:

$$f(x)=\sum^5_{i=1} i \cos\left[\left(i+1\right)x+i\right]$$

where:

\(\bullet\) \(-10 \leq x \leq 10\)

\(\bullet\) It has three global maximum: \(f_{max}(x^*)=14.508007927195038\)

\(\bullet\) \(x^*=\{-7.08350640682890, -0.800321099691209, 5.48286420658132\}\) (determined by us using MapleSoft 2015)

\(\bullet\) This function is similar to Problem No.3; and also it comes with different names, like: Lévy's Function No.2 [1], Modified Trigonometric Polynomial Function [2] and Problem No.8 [3].

\(\bullet\) It has to be known that the original function in [1] comes with minimization mode (i.e., \(f_{original}=-f_{used}\)). The given expression here is similar to that used in [3]; so it becomes suited for maximization algorithms.

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 # 8 (Modified Trigonometric Polynomial or Lévy's Function # 2)
% Range of initial points: -10 <= x <= 10
% It has three global maximum: x={-7.08350640682890, -0.800321099691209,
% 5.48286420658132}
% f(x)=14.508007927195038
% Coded by: Ali R. Alroomi | Last Update: 25 Feb. 2015 | www.al-roomi.org
 
clear
clc
warning off
 
xmin=-10;
xmax=10;
R=100000; % steps resolution
x=xmin:(xmax-xmin)/R:xmax;
 
for i=1:length(x)
    
    for j=1:5
        fs(j)=j*cos((j+1)*x(i)+j);
    end
    
    f(i)=sum(fs);
    
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. Lévy, 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] Bruno O. Shubert, "A Sequential Method Seeking the Global Maximum of a Function," SIAM Journal of Numerical Analysis, vol. 9, no. 3, pp. 379-388, 1972.
[3] 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.
[4] 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