Problem No.03 (Shubert's, Trigonometric Polynomial or Suharev-Zilinskas' Function)

I. Mathematical Expression:

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

where:

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

\(\bullet\) It has three global maximum: \(f_{max}(x^*)=12.031249442167146\)

\(\bullet\) \(x^*= \{-6.77457614347361, -0.491390835930674, 5.79179447080188\}\) (determined by us using MapleSoft 2015)

\(\bullet\) This function is known with different names, like: Shubert's Function, Trigonometric Polynomial Function [1], Problem No.3 [2] and Suharev-Zilinskas' Function [3].

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 # 3 (Shubert's, Trigonometric Polynomial, or Suharev-Zilinskas' Function)
% Range of initial points: -10 <= x <= 10
% It has three global maximum: x={-6.77457614347361, -0.491390835930674,
% 5.79179447080188}
% f(x)=12.031249442167146
% Coded by: Ali R. Alroomi | Last Update: 04 March 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*sin((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] 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.
[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] E. Kiseleva and T. Stepanchuk, "On the Efficiency of a Global Non-Differentiable Optimization Algorithm Based on the Method of Optimal Set Partitioning," Journal of Global Optimization, vol. 25, no. 2, pp. 209-235, Feb. 2003.
[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