Week 4.1 - Genetic Algorithm

% Stalagmite Mathematical Model for optimization

clear all
close all
clc

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


% Evaluating the Stalagmite function
for i=1:length(x)
    for j = 1:length(y)
        input_vector(1)= x(i);
        input_vector(2)= y(j);
        f(i,j) = stalagmite(input_vector);
    end
end

surf(x,y,f)
xlabel('x value')
ylabel('y value')

shading interp
[input, fval]=ga(@stalagmite,2)

STEPS

1. firstly we use linspace command to create 150 values from 0 to 0.6 .

2. We use meshgrid command to evaluate the functions of two variables and three dimensional plots

3. We use the stalagmite function for different iterations .

4. We use the ga function to define the stalagmite function for various iterations .

5. We measure the time for each of 50 iterations and the optimum value is obtained .

Genetic Algorithm:

Genetic Algorithm ( GA ) is a branch of field of study called evolutionary comutation . It is based on evolution theory by Darwin . GA is used to optimize a given function to find its maximum or minimum value .

The basic components of a GA are-

1. fitness function : It is the function that GA is trying to optimize

2. a population of chromosome : It is the numerical values that represent all the solutions to the function .The selection operator chooses some of the chromosomes ( with replacement ) for reproduction based on probability distribution . fitter the chromosome more likely it is to be selected .

3. cross over : This operator swaps in random a subsequence of two of the chosen chromosomes to create two offspring . Usually each chromosome is converted into binary code and then cross over is done from one parent chromosome into another .

4. Mutation : Mutation operator randomly flips individual bits in the new chromosomes .

The GA syntax in Matlab is given by,

x =ga(fun,nvars,A,B.Aeq.BeqLB,UB.nonicon,intcon.options)

• fun = objective function
• nvars = number of variables
• A and B = subject to linear inequalities of the form A * x < = B
• Aeq and Beq = subject to linear equalities of the form A * x = B
• LB = lower bound
• UB = upper bound
• nonicon = non linear inequalities
• intcon = integer constrained
• options = helps to add contraint tolerance by increasing the population size

% Stalagmite Function

function f= stalagmite(input_vector)
x=input_vector(1);
y=input_vector(2);
f1x=(sin((5.1*pi*x)+0.5)).^6;
f2y=(sin((5.1*pi*y)+0.5)).^6;
f3x=exp(-4*log(2)*(x-0.0667).^2/0.64);
f4y=exp(-4*log(2)*(y-0.0667).^2/0.64);
f = -(f1x*f2y*f3x*f4y);
end

% Stalagmite Mathematical Model for optimization

clear all
close all
clc

% Defining our search space
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);
num_cases=50

% Evaluating the Stalagmite function
for i=1:length(x)
    for j = 1:length(y)
        input_vector(1)= x(i);
        input_vector(2)= y(j);
        f(i,j) = stalagmite(input_vector);
    end
end

% study 1 unbounded plot

for k =1:num_cases
    [ip1 fopt1(k)]=ga(@stalagmite,2) 
    xopt1(k)= ip1(1) 
    yopt1(k)= ip1(2) 
end

figure(1) 
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp

plot3( xopt1,yopt1,-fopt1,'marker','o','markerFaceColor','r') 
xlabel('x data') 
ylabel('y data')
title ('unbounded plot') 
subplot(2,1,2)
plot(-fopt1)
xlabel('iterations') 
ylabel('fopt1')
title ('unbounded plot')


% study 2 bounded plot

for k = 1 : num_cases
    [ip2 fopt2(k)] = ga(@stalagmite,2,[],[],[],[],[0;0],[0.6;0.6]);
    xopt2(k) = ip2(1)
    yopt2(k) = ip2(2) 
end 

figure(2) 
subplot(2,1,1) 
hold on
surfc(x,y,-f) 
shading interp 

plot3(xopt2,yopt2,-fopt2,'marker','o','markerfacecolor','r')
xlabel('x data') 
ylabel('y data')
title ('bounded plot') 
subplot(2,1,2)
plot (-fopt2) 
xlabel('iterations') 
ylabel('fopt1')
title ('bounded plot')

% study 3 bounded plot with population
options =optimoptions('ga');
options =optimoptions(options,'populationsize',500); 

for k = 1:num_cases
    [ip3, fopt3(k)]=ga(@stalagmite,2,[],[],[],[],[0;0],[0.6;0.6],[],[],options); 
    xopt3(k) = ip3(1); 
    yopt3(k) = ip3(2); 
end 

figure(3)
subplot(2,1,1) 
hold on 
surfc(x,y,-f)
shading interp 
plot3(xopt3,yopt3,-fopt3,'marker','o','markerfacecolor','r') 
xlabel('x data') 
ylabel('y data')
title('Bounded plot with population')

subplot(2,1,2)
plot(-fopt3) 
xlabel('iterations') 
ylabel('fopt1')
title('Bounded plot with population')