Week 4.1 - Genetic Algorithm

% Evaluating stalagmite function using Genetic algorithm.



clear all
close all
clc

%defining value of x and y,taking 150 values between 0 to 0.6
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
num_cases = 50

%creating 2D mesh using mesh grid command.
[xx,yy] = meshgrid(x,y);

% Evaluating Stalagmite 150 iterations.

for i= 1:length(x)
    for j= 1: length(y) 
        input_vector(1)=xx(i,j);
        input_vector(2)=yy(i,j);
        d(i,j)= stalagmite(input_vector);
    end
end  
%study 1 - Stastistical behaviour by using genetic algorithm by taking 50 iteration and considering
%only 2 variables 
tic
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,d)
shading interp
plot3(xopt,yopt,fopt,'marker', 'o', 'markersize',5)
title('unbounded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('iterations')
ylabel('Function minimum')

%study 2 - Statistical behavior by setting an upper and lower limits.
tic
for i= 1: num_cases
    [inputs, fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]);
        xopt(i)= inputs(1);
        yopt(i)= inputs(2);
end
study2_time = toc;

figure(2)
subplot(2,1,1)
hold on
surfc(x,y,d)
shading interp
plot3(xopt,yopt,fopt,'marker', 'o', 'markersize',5)
title('unbounded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('iterations')
ylabel('Function minimum')

%study 3 - By increasing GA iterations byh using optimoptions.
options = optimoptions('ga');
options = optimoptions(options,'populationsize',170);

tic
for i= 1: num_cases
    [inputs, fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);
        xopt(i)= inputs(1);
        yopt(i)= inputs(2);
end
study3_time = toc;

figure(3)
subplot(2,1,1)
hold on
surfc(x,y,d)
shading interp
plot3(xopt,yopt,fopt,'marker', 'o', 'markersize',5)
title('unbounded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('iterations')
ylabel('Function minimum')

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

%Stalagmite function code

%here we take the output as -f, As we need minimum values.

function f = stalagmite(input_vector)
% 
x= input_vector(1);
y= input_vector(2);

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

f =(-f1*f2*f3*f4);

end

Explanation

Theory about GA:

A genetic algorithm (GA) is a method for solving both constrained and unconstrained optimization problems based on a natural selection process that mimics biological evolution.

Five phases are considered in a genetic algorithm.

1. Initial population
2. Fitness function
3. Selection
4. Crossover
5. Mutation

Genetic Algorithms - Termination Condition

1. When there has been no improvement in the population for X iterations.
2. When we reach an absolute number of generations.
3. When the objective function value has reached a certain pre-defined value.

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

Significations of running multiple iterations:

To get accurate solutions

Result:

All the plots attached contain 2 graph

1) unbounded graph 3D plot. which statistically represents stalagmite functions.

2) Iterations v/s minimum function value.

Study 1: Shows the plot using the genetic algorithm by taking 50 iterations and considering only 2 variables. This value varies from -1 to 0. Which has a huge range of minimal values and needs more optimization.

Study 2: This study is improvised by applying lower and upper limit bound. This value varies from -0.4 to -1 . and the Values obtained are more. This study gives little more stabilized output but still requires optimization

Study 3: This study is improvised by increasing the number of iterations by using 'optimoptions' function. This gives a very much stabilized graph. and there are only 2 minimal values.