GLOBAL MAXIMA BY GENETIC ALGORITHM

AIM:To evaluate the global maxima by genetic algorithm using matlab.

THEORY:

Genetic algorithm:ga is the solver like ode 45 but ga optimizes the function where as ode 45 solve ode.In matlab,ga is built only to minimize value of the function.

Optimization:It is to maximize or minimize the function.

Crossover:It is a convergence operation which is intended to pull the population towards a local maximum/minimum.

Mutation:it is intended to break one or more members of population out of local maximum/minimum.

SYNTAX:

1.[x,fval]=ga(fun,nvars)

where,

x=inputs

fval=function value

ga=genetic

fun=function name

nvars=number of variables

2.[x,fval]=ga(fun,nvars,A,b,Aeq,beq,lb,ub)

where,

A,b,Aeq,beq=defines linear and non-linear constraints

lb=lower bounds

ub=upper bounds

3.[x,fval]=ga(fun,nvars,A,b,[],[],nonlcon,intcon,options)

where,

nonclon=non-linear constraints

intcon=integer constraints

options=optimisation option

4.options=optimoptions(solvername)

solver name=ga

5.options=optimoptions(solvername,name,value)

where,

name=population size

value=populationsize value

FUNCTION CODE:

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

MAIN CODE:

clear all
close all
clc

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

% creating a 2d 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(input_vector);
    end
end

surfc(xx,yy,-f)
xlabel('x value')
ylabel('y value')

shading interp

[inputs ,fval]=ga(@stalagmite,2)

FIGURE(1):

CODING FOR ITERATIONS:

clear all
close all
clc

x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
num_cases=50
% creating a 2d 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(input_vector);
    end
end


tic
% study 1-satistical behaviour
for i=1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
study1_time=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 inputs')
subplot(2,1,2)
plot(-fopt)
xlabel('Iterations')
ylabel('function minimum')
tic;



% study 2 satistical behaviour 
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_time=toc

figure(2)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('bounded inputs')
subplot(2,1,2)
plot(-fopt)
xlabel('Iterations')
ylabel('function maximum')

% increasing ga iterations
options=optimoptions(@ga)
options=optimoptions(@ga,'populationsize',50)

tic;


% study 3 satistical behaviour
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
study3_time=toc

figure(3)

subplot(2,1,1)
hold on
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',5,'markerfacecolor','r')

subplot(2,1,2)
plot(-fopt)
xlabel('Iterations')
ylabel('function maximum')
tic;



    

CONCLUSION:

surface plot generated by ga for unbounded inputs:

surface plot generated by ga for bounded inputs:

surface plot generated by ga with lower and upper bounds and increased population size: