Price's Function No.03 (Modified Rosenbrock's or Price-Rosenbrock's Function)

I. Mathematical Expression:

$$f(X)=100\left(x_2-x_1^2\right)^2+\left[6.4\left(x_2-0.5\right)^2-x_1-0.6\right]^2$$

where:

\(\bullet\) \(-5\leq x_i\leq 5\) , \(i=1,2\) 

\(\bullet\) It has two global minimum: \(f_{min}(X^*)=0\)

\(\bullet\) \(x^*_i =\{(0.341307503353524, 0.116490811845416), (1, 1)\}\)

\(\bullet\) This function was designed, by W. L. Price in [1], to assess the performance of the Controlled Random Search Optimization Algorithm when the minima lie within a steep-sided, nonlinear valley.

\(\bullet\) It is called Modified Rosenbrock's Function [1]. Also, it is known as Price's Function No.03 [2, 3]. Because this function is based on Rosenbrock's Valley Function and there are many other modified versions, so it is better to call this function as Price-Rosenbrock's Function [4].

\(\bullet\) The two optimum points are corresponded to the intersections of the two parabolas. A local minimum with \(f<10^{-2}\) is found when the parabolas approach without intersection [1].

\(\bullet\) There are some typoerrors in [2, 3]: they mention four global minimum "the optimal points of Price's Function No.01", which is wrong. Also, two different side constraints are considered; \(X\in[-500,500]\) as in [2] and \(X\in[-50,50]\) as in [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. 2&3D-Plots:

IV. Controllable 3D Model:

- In case you want to adjust the rendering mode, camera position, background color or/and 3D measurement tool, please check the following link

- In case you face any problem to run this model on your internet browser (it does not work on mobile phones), please check the following link

V. MATLAB M-File:

% Price's Function # 3 (Modified Rosenbrock's or Price-Rosenbrock's Function)
% Range of initial points: -5 <= xj <= 5 , j=1,2 
% Some references consider: -50 <= xj <= 50 or -500 <= xj <= 500
% Two global minimum: (x1,x2)={(0.341307503353524,0.116490811845416),(1,1)}
% f(x1,x2)=0
% Coded by: Ali R. Alroomi | Last Update: 14 June 2015 | www.al-roomi.org

% It is based on the Rosenbrock's Function

clear
clc
warning off

x1min=-5;
x1max=5;
x2min=-5;
x2max=5;
R=1500; % steps resolution
x1=x1min:(x1max-x1min)/R:x1max;
x2=x2min:(x2max-x2min)/R:x2max;

for j=1:length(x1)
    
    for i=1:length(x2)
        f(i)=100*(x2(i)-x1(j).^2).^2+(6.4*(x2(i)-0.5).^2-x1(j)-0.6).^2;
    end

    f_tot(j,:)=f;

end

figure(1)
meshc(x1,x2,f_tot);colorbar;set(gca,'FontSize',12);shading interp;
xlabel('x_2','FontName','Times','FontSize',20,'FontAngle','italic');
set(get(gca,'xlabel'),'rotation',25,'VerticalAlignment','bottom');
ylabel('x_1','FontName','Times','FontSize',20,'FontAngle','italic');
set(get(gca,'ylabel'),'rotation',-25,'VerticalAlignment','bottom');
zlabel('f(X)','FontName','Times','FontSize',20,'FontAngle','italic');
title('3D View','FontName','Times','FontSize',24,'FontWeight','bold');

figure(2)
mesh(x1,x2,f_tot);view(0,90);colorbar;set(gca,'FontSize',12);
xlabel('x_2','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('x_1','FontName','Times','FontSize',20,'FontAngle','italic');
zlabel('f(X)','FontName','Times','FontSize',20,'FontAngle','italic');
title('X-Y Plane View','FontName','Times','FontSize',24,'FontWeight','bold');

figure(3)
mesh(x1,x2,f_tot);view(90,0);colorbar;set(gca,'FontSize',12);
xlabel('x_2','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('x_1','FontName','Times','FontSize',20,'FontAngle','italic');
zlabel('f(X)','FontName','Times','FontSize',20,'FontAngle','italic');
title('X-Z Plane View','FontName','Times','FontSize',24,'FontWeight','bold');

figure(4)
mesh(x1,x2,f_tot);view(0,0);colorbar;set(gca,'FontSize',12);
xlabel('x_2','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('x_1','FontName','Times','FontSize',20,'FontAngle','italic');
zlabel('f(X)','FontName','Times','FontSize',20,'FontAngle','italic');
title('Y-Z Plane View','FontName','Times','FontSize',24,'FontWeight','bold');

Click here to download m-file

VI. References:

[1] W. L. Price, “A Controlled Random Search Procedure for Global Optimisation,” The Computer Journal, vol. 20, no. 4, pp. 367-370, 1977. [Online]. Available: http://comjnl.oxfordjournals.org/content/20/4/367.full.pdf
[2] M. Jamil and X. S. Yang, "A Literature Survey of Benchmark Functions for Global Optimization Problems," International Journal of Mathematical Modelling and Numerical Optimisation, vol. 4, no. 2, pp. 150–194, Aug. 2013.
[3] A. Gavana, "Test Functions Index," Feb. 2013, [Accessed April 01, 2013]. [Online]. Available: http://infinity77.net/global_optimization/test_functions.html
[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