Genetic Algorithm

AIM: To write a code to optimise stalagmite function and to find the global maxima of the function.

THEORY: 

 Genetic algorithm (ga) is a method of solving both constrained and uncontrained optimization problems based on natural selection that mimics biological evolution.

ga is the solver like ode45 but it optimize the function where as ode45 solve ODE.

In matlab ga is built only to find out the minimum value of  the function.

Parameters of genetic algorithm:

Selection:The process  that determines which solutions are to be preserved and allowed to reproduce                   and which ones deserve to die out.

Crossover:It is a convergence operation which is used to pulled out population towards maxima or                        minima.

Mutation:It is a gentic operator used to maintain genetic diversity from one generation of a genetic                       alogorithm (example chromosomes) to the next.

ga syntax:

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

   fun=function name 

   nvars=number of varibles

   A and b =linear inequality constraints

   lb=lower bounds 

   ub=upper bounds

   nonlcon=non linear constraints

   options=optimization options.

Function code used :

function[inputs]=stalagmite(input_vector)
x1=input_vector(1);
y1=input_vector(2);
f1_x=(sin(5.1*pi*x1+0.5))^6;

f1_y=(sin(5.1*pi*y1+0.5))^6;

f2_x=exp(-4*log(2)*(x1-0.0667)^2/0.64);

f2_y=exp(-4*log(2)*(y1-0.0667)^2/0.64);

[inputs]=-(f1_x.*f1_y.*f2_x.*f2_y);

end

Code for global maxima

clear all
close all 
clc


%defining our search space 

x=linspace(0,0.6,150);
y=linspace(0,0.6,150);




%creating 2d mesh
[xx yy]=meshgrid(x,y);

%stalagmaite function evaualting
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
surfc(xx,yy,f)
xlabel('x value')
ylabel('y value')
shading interp
[inputs, fval]=ga(@stalagmite,2)

To find out minimum value of the function ga

clear all
close all 
clc


x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
num_cases=50
[xx yy]=meshgrid(x,y);

%stalagmaite function evaualting
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



surf(xx,yy,-f)
shading interp
[inputs, fval]=ga(@stalagmite,2)

xlabel('x value')
ylabel('y value')
title('staglamite function')



tic



% study statistical behaviour

for i=1:num_cases
    [inputs, fopt(i)]=ga(@stalagmite, 2);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
study1=toc

figure(2)


subplot(3,1,1)
hold on
surfc(x,y,-f)
title('unbounded inputs')
xlabel('x value')
ylabel('y value')

shading interp
hold on 
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')


subplot(3,1,2)
plot(-fopt)
subplot(3,1,3)
plot(-fopt)

xlabel('iterations')
ylabel('function maximum')
tic;




for i=1:num_cases
    [inputs, fopt(i)]=ga(@stalagmite, 2,[],[],[],[],[0;0],[0.07;0.07]);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end



study2=toc
figure(3)
subplot(2,1,1)

surfc(x,y,-f)
shading interp
title('bounded inputs')
xlabel('x value')
ylabel('y value')
hold on

plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')


subplot(2,1,2)
plot(-fopt)

xlabel('iterations')
ylabel('function maximum')

options=optimoptions(@ga)
options=optimoptions(@ga,'populationsize',300)
tic

for i=1:num_cases
    [inputs, fopt(i)]=ga(@stalagmite, 2,[],[],[],[],[0;0],[0.07;0.07],[],[],options);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end

study3time=toc
figure(4)

subplot(2,1,1)
surfc(x,y,-f)
shading interp
title('bounded inputs with increased population size')

xlabel('x value')
ylabel('y value')
hold on

plot3(xopt,yopt,-fopt,'marker','o','markersize',4,'markerfacecolor','r')

subplot(2,1,2)
plot(-fopt)

xlabel('iterations')
ylabel('function maximum')
tic;

 Staglamite function:

function with ga

Results:

       Minimum value of the function is find out and the global maxima for 3studies  are showed in the           above graphs.