Optimization and FInding the Global maxima of a stalagmite function

• Selection rules select the individuals, called parents, that contribute to the population at the next generation.

• Crossover rules combine two parents to form children for the next generation.

• Mutation rules apply random changes to individual parents to form children.

Syntax for Genetic Algorithm:-

x = ga(fun,nvars,A,b,[],[],lb,ub,nonlcon,IntCon,options)

Code according to the given challenge:-

%code to optimize the stalagmite function and finding the global maxima of
%the function
clear all
close all
clc

%search space
x= linspace(0,0.6,150);
y= linspace(0,0.6,150);

%2D meshgrid
[xx yy]= meshgrid(x,y);

num_case=150;

%looping

for i=1:length(xx)
for j=1:length(yy)
    
    input_vector(1)=xx(i,j);
    input_vector(2)=yy(i,j);
    F(i,j)= stalagmite(input_vector);
    
end
end

%case 1:: Unbounded values
tic
for i=1:num_case
    
    [inputs,Fopt1(i)]= ga(@stalagmite,2);
  xopt(i)= inputs(1);
  yopt(j)= inputs(2);
end

study_time_1=toc;

%plotting
 figure(1)
 subplot(2,1,1);
 hold on
 surfc(xx,yy,F);
 shading interp;
 ylabel(\'Y data\');
 xlabel(\'X data\');
 plot3(xopt,yopt,Fopt1,\'marker\',\'o\',\'markersize\',5);
 title(\'unbounded inputs\')
 subplot(2,1,2)
 plot(Fopt1)
 xlabel(\'Iterations\')
 ylabel(\'Function Maximum\')
 
%case 2 Bounding the values
tic
for i=1:num_case
    
   [inputs,Fopt2(i)]= ga(@stalagmite,2,[],[],[],[],[0:0],[1:1]);  
  xopt(i)= inputs(1);
  yopt(j)= inputs(2);
end

study_time_2=toc;

%plotting
 figure(2)
 subplot(2,1,1);
 hold on
 surfc(xx,yy,F);
 shading interp;
 ylabel(\'Y data\');
 xlabel(\'X data\');
 plot3(xopt,yopt,Fopt2,\'marker\',\'o\',\'markersize\',5);
 title(\'bounded inputs\')
 subplot(2,1,2)
 plot(Fopt2)
 xlabel(\'Iterations\')
 ylabel(\'Function Maximum\')
 
 %case 3:: bounding the values and increasing population size by 500
 
 %for creating optimization options
 options= optimoptions(\'ga\');
 options= optimoptions(options,\'populationsize\',500);
 
  tic   %time variable
for i=1:num_case
    
   [inputs,Fopt3(i)]= ga(@stalagmite,2,[],[],[],[],[0:0],[1:1],[],[],options);  
  xopt(i)= inputs(1);
  yopt(j)= inputs(2);
end

study_time_3=toc;

%plotting
 figure(3)
 subplot(2,1,1);
 hold on
 surfc(xx,yy,F);
 shading interp;
 ylabel(\'Y data\');
 xlabel(\'X data\');
 plot3(xopt,yopt,Fopt3,\'marker\',\'o\',\'markersize\',5);
 title(\'bounded inputs with increasing population size\')
 subplot(2,1,2)
 plot(Fopt3)
 xlabel(\'Iterations\')
 ylabel(\'Function Maximum\')
 
 max_value=(Fopt3)
 

%code for stalagmite function:::

function [F]= stalagmite(input_vector)

x=input_vector(1);
y=input_vector(2);
f1=sin(5.1*pi*x + 0.5).^6;
f3=exp(-4*log(2)*(x-0.0667)^2)/0.64;
f2=sin(5.1*pi*y + 0.5).^6;
f4=exp(-4*log(2)*(y-0.0667)^2)/0.64;
F=(f1*f2*f3*f4);
F=1/(F+1);
end

Outputs:-

time for the 3 cases:- 30.1794,33.9548 and 158.7010.