- Parent Category: Unconstrained
- Category: n-Dimensions
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Type-II Medium-Complex Deceptive Problem
I. Mathematical Expression:
$$f(X)=\Bigg[\frac{\displaystyle 1}{\displaystyle n} \sum^n_{i=1}g_i(x_i)\Bigg]^{\beta}$$
where:
\(\bullet\) \(\beta\) is a non-linearity factor. It is considered equal to 1 in [1]
\(\bullet\) For each dimension \(i\), \(g_i(x_i)\) is randomly defined from the two main-cases and two sub-cases as follows:
\(\bullet\) \(g_i(x_i)=\begin{cases}
\begin{cases}
\alpha-x_i & \text{ if } 0 \leq x_i \leq \alpha \\
\frac{\displaystyle x_i-\alpha}{\displaystyle 1-\alpha} & \text{ otherwise }
\end{cases} & \text{ if round} \left(rand_i\right)=0 \\
\begin{cases}
\frac{\displaystyle 1-\alpha-x_i}{\displaystyle 1-\alpha} & \text{ if } 0 \leq x_i < 1-\alpha \\
x_i-1+\alpha & \text{ otherwise }
\end{cases} & \text{ if round} \left(rand_i\right)=1
\end{cases}\)
\(\bullet\) \(\alpha=0.8\)
\(\bullet\) \(0\leq x_i\leq 1\) , \(i=1,2,\cdots,n\)
\(\bullet\) It has two global maximum: \(f_{max}(X^*)=1\) with \(2^n-1\) local maximum
\(\bullet\) \(x^*_i=\{0,1\}\)
\(\bullet\) For \(n=1\), there are two possibilities for \(f(x)\), because it is constructed by one \(g(x)\). While for \(n=2\), there are four possibilities for \(f(X)\), because it depends on the summation of \(g_1(x_1)\) and \(g_2(x_2)\).
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:
\(\bullet\) If \(rand_1=0\) and \(rand_2=0\):
\(\bullet\) If \(rand_1=1\) and \(rand_2=0\):
\(\bullet\) If \(rand_1=0\) and \(rand_2=1\):
\(\bullet\) If \(rand_1=1\) and \(rand_2=1\):
IV. Controllable 3D Model:
\(\bullet\) The following model is constructed based on the first possibility \(\rightarrow\) \(rand_1=0\) and \(rand_2=0\):
- 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:
% Type-II Medium-Complex Deceptive Problem
% Range of initial points: 0 <= xj <= 1 , j=1,2,...,n
% Two global maximum: (x1,x2,...,xn)={0,1} "with (2^n)-1 local maximum"
% f(X)=1
% Coded by: Ali R. Alroomi | Last Update: 27 July 2015 | www.al-roomi.org
clear
clc
warning off
alpha=0.8;
beta=1;
Select1=round(rand);
Select2=round(rand);
x1min=0;
x1max=1;
x2min=0;
x2max=1;
R=1500; % steps resolution
x1=x1min:(x1max-x1min)/R:x1max;
x2=x2min:(x2max-x2min)/R:x2max;
for j=1:length(x1)
% For 1-dimensional plotting
if Select1==0
if x1(j)<=alpha
g1=alpha-x1(j);
else
g1=(x1(j)-alpha)/(1-alpha);
end
else
if x1(j)<(1-alpha)
g1=(1-alpha-x1(j))/(1-alpha);
else
g1=x1(j)-1+alpha;
end
end
f1(j)=g1^beta;
% For 2-dimensional plotting
if Select2==0
for i=1:length(x2)
if x2(i)<=alpha
g2=alpha-x2(i);
else
g2=(x2(i)-alpha)/(1-alpha);
end
fn(i)=(0.5*(g1+g2))^beta;
end
else
for i=1:length(x2)
if x2(i)<(1-alpha)
g2=(1-alpha-x2(i))/(1-alpha);
else
g2=x2(i)-1+alpha;
end
fn(i)=(0.5*(g1+g2))^beta;
end
end
fn_tot(j,:)=fn;
end
figure(1)
plot(x1,f1,'r','LineWidth',2);set(gca,'FontSize',12);
xlabel('x','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('f(x)','FontName','Times','FontSize',20,'FontAngle','italic');
title('2D View','FontName','Times','FontSize',24,'FontWeight','bold');
figure(2)
meshc(x1,x2,fn_tot);colorbar;set(gca,'FontSize',12);
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(3)
mesh(x1,x2,fn_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(4)
mesh(x1,x2,fn_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(5)
mesh(x1,x2,fn_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');
VI. References:
[1] Hideaki Suzuki, and Hidefumi Sawai, "Chemical Genetic Algorithms --- Coevolution between Codes and Code Translation," in Proceedings of the Eighth International Conference on Artificial Life (Artificial Life VIII), 2002, pp. 164-172. [Online]. Available: http://www.alife.org/alife8/proceedings/sub2029.pdf
[2] 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
