Problem No.10 (Fichtenholz's Function No.01 or Himmelblau's Function No.02)

I. Mathematical Expression:

$$f(x)=x \sin(x)$$

where:

\(\rhd\) For Fichtenholz's Function No.1 (Problem No.10) [1, 2]:

\(\bullet\) \(0 \leq x \leq 10\)

\(\bullet\) It has one global maxima: \(f_{max}(x^*)=7.916727371587783\)

\(\bullet\) \(x^*=7.978665706811959\) (determined by Maple 2015)

\(\rhd\) For Himmelblau's Function No.2 [3]:

\(\bullet\) \(-7.85 \leq x \leq 7.85\)

\(\bullet\) It has two global minimum: \(f_{min}(x^*)=-4.814469889712269\)

\(\bullet\) \(x^*=\{-4.913180450455435, 4.913180450455435\}\) (determined by Maple 2015)

\(\bullet\) Based on the above information, this function has different names, like: Fichtenholz's Function No.1 [1], Problem No.9  [2] and Himmelblau's Function No.2 (he considered the global minimum points with the same function expression; i.e., without using \(f_{Himmelblau}=-f_{Fichtenholz}\)) [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:

\(\rhd\) For Fichtenholz's Function No.1 (Problem No.10) [1, 2]:

\(\rhd\) For Himmelblau's Function No.2 [3]:

IV. MATLAB M-File:

% Problem # 10 (Fichtenholz's Function # 1 or Himmelblau's Function # 2)
%
% For Fichtenholz's Function No.1 (Problem # 10):
% Range of initial points: 0 <= x <= 10 (see for Himmelblau)
% Global maxima: x=7.978665706811959
% f(x)=7.916727371587783
%
% For Himmelblau's Function No.2:
% Range of initial points: -7.85 <= x <= 7.85
% Two global minimum: x={-4.913180450455435,4.913180450455435}
% f(x)=-4.814469889712269
% Coded by: Ali R. Alroomi | Last Update: 14 Feb. 2015 | www.al-roomi.org
 
clear
clc
warning off
 
S=2; % to select between Fichtenholz and Himmelblau 
if S==1 % Fichtenholz
    xmin=0;
    xmax=10;
else % Himmelblau
    xmin=-7.85;
    xmax=7.85;
end
R=100000; % steps resolution
x=xmin:(xmax-xmin)/R:xmax;
 
for i=1:length(x)
    f(i)=x(i)*sin(x(i));
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] G.M. Fichtenholz, Differential, und lntegralrechnung I. Berlin: Deutscher Verlag der Wissenschaften, 1964.
[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