Problem No.06 (Zilinskas-Shaltyanis' or Richard Brent's Function No.01)

I. Mathematical Expression:

$$f(x)=\left[x+\sin(x)\right]e^{-x^2}$$

where:

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

\(\bullet\) \(f_{max}(x^*)=0.824239398476077\)

\(\bullet\) \(x^*=0.679578666600993\) (determined by us using MapleSoft 2015)

\(\bullet\) This function is known with different names, like: Richard Brent's Function No.1 [1] (also there is a 2-dimensional function called Brent's Function), Problem No.6 [2] and Zilinskas-Shaltyanis' 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 # 6 (Zilinskas-Shaltyanis' or Richard Brent's Function # 1)
% Range of initial points: -10 <= x <= 10
% Global maxima: x=0.679578666600993 
% f(x)=0.824239398476077
% Coded by: Ali R. Alroomi | Last Update: 14 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)
    f(i)=(x(i)+sin(x(i)))*exp(-x(i)^2);
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] R.P. Brent, Algorithms for Minimization without Derivatives. Englewood Cliffs, New Jersey: Prentice-Hall, 1973.
[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