Genetic Algorithm

x = ga(fun,nvars,A,b) finds a local minimum x to fun, subject to the linear inequalities A*x ≤ b. ga evaluates the matrix product A*x as if x is transposed (A*x').

x = ga(fun,nvars,A,b,Aeq,beq) finds a local minimum x to fun, subject to the linear equalities Aeq*x = beq and A*x ≤ b. (Set A=[] and b=[] if no linear inequalities exist.) ga evaluates the matrix product Aeq*x as if x is transposed (Aeq*x').

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that a solution is found in the range lb ≤ x ≤ ub. (Set Aeq=[] and beq=[] if no linear equalities exist.)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon) subjects the minimization to the constraints defined in nonlcon. The function nonlcon accepts x and returns vectors C and Ceq, representing the nonlinear inequalities and equalities respectively. ga minimizes the fun such that C(x) ≤ 0 and Ceq(x) = 0. (Set lb=[] and ub=[] if no bounds exist.)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options) minimizes with the default optimization parameters replaced by values in options. (Set nonlcon=[] if no nonlinear constraints exist.) Create options using optimoptions.

[x,fval,exitflag,output,population,scores] = ga(___) also returns a matrix population, whose rows are the final population, and a vector scores, the scores of the final population.

Program Code:

clear all
close all
clc

%Defining our search space 
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);
num_spaces=50;
%Creating a 2Dimensional mesh
[xx,yy]= meshgrid(x,y);

%Evaluating the stalagmite function
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_func(input_vector);
end
end
%Study1-Statistical behavior-unbounded
tic
for i= 1:num_spaces
    [results,fopt(i)]= ga(@stalagmite_func,2);
    xopt(i)=results(1);
    yopt(i)=results(2);
end
study_time1=toc;
figure(1)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('unbounded input')
subplot(2,1,2)
plot(-fopt)
xlabel('iteration')
ylabel('function maxima')
tic
%Study2-Statistical behavior-with upper and lower bounds
for i = 1:num_spaces
    [results,fopt(i)]= ga(@stalagmite_func,2,[],[],[],[],[0;0],[0.6;0.6]);
    xopt(i)=results(1);
    yopt(i)=results(2);
end
study_time2=toc;
figure(2)
subplot(2,1,1)
hold on
surfc(xx,yy,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('bounded input')
subplot(2,1,2)
plot(-fopt)
xlabel('iteration')
ylabel('function maxima')

%Study3 increasing GA iterations
options = optimoptions('ga');
options = optimoptions(options,'PopulationSize',150);
tic
for i = 1:num_spaces
    [results,fopt(i)]= ga(@stalagmite_func,2,[],[],[],[],[0;0],[0.6;0.6],[],[],options);
    xopt(i)=results(1);
    yopt(i)=results(2);
end
study_time3=toc;
figure(3)
subplot(2,1,1)
hold on
surfc(xx,yy,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('bounded input with modified population size')
subplot(2,1,2)
plot(-fopt)
xlabel('iteration')
ylabel('function maxima')

Stalagmite Function code:

function [f]=stalagmite_func(input_vector)

x=input_vector(1);
y=input_vector(2);
term1=(sin(5.1*pi *x +0.5)).^6;
term2=(sin(5.1*pi *y +0.5)).^6;
term3=exp((-4*0.6931).*(x-0.0667)^2/(0.64));
term4=exp((-4*0.6931).*(y-0.0667)^2/(0.64));

f= -(term1.*term2.*term3.*term4);
end