Week 4.1 - Genetic Algorithm

STALAGMITE FUNCTION

AIM:

To write the code to calculate the global maxima of the stalagmite function and to explain the syntax for genetic algoritham(ga) in MATLAB

OBJECTIVE:

• To write a code in MATLAB to optimize the stalagmite function and find global maxima of the function.
• To explain the concept of genetic algoritham and the syntax for ga in MATLAB.
• To plot the graph of all three studies and also for global maxima.

THEORY:

A stalagmite is a type of rock that rises from the floor of a cave due to the accumulation of materials decomposed on the floor from ceiling drippings. That is why we are using the stalagmite as the name for the program due to its shape.

Genetic Algoritham(ga)

A genetic algoritham is an algoritham that imitates the process of natural selection. They help solve optimisation and search problems.Genetic algorithams Genetic algoritham imitates natural biological process such as inheritance,mutation,selection and crossover.The concept of algoritham is a search technique often used compute science to find complex non-obvious solution to algorithamic optimisation and search problems.

MATHEMATICAL EQUATION:

To find global maxima

f = (f1.f2.f3.f4)

To find global minima

f = (f1.f2.f3.f4)

where

f1 = [sin(5.1pix+0.5)]^6

f2 = [sin(5.1pix+0.5]^6

f3 = exp[(-4log(2)*((x-0.0667)^2)/0.64))]

f4 = exp[(-4log(2)*((y-0.0667)^2)/0.64))]

here x and y are two variables.

PROGRAM:

clear all
close all
clc

% defining the 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 stalgamite 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)

FUNCTION:

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

Here for ploting global minima and finding the minimum of the function ga is used

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 stalgamite 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

% Study1 - statistical behavior

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

study1_time = toc;

% plotting1
figure(1)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','O','markersize',5,'markerfacecolor','g')
title('Unbounded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('function minimum')

tic
% study 2- statistical behaviour
for i = 1:num_cases
    [inputs,fopt(i)] = ga(@stalagmite,2,[],[],[],[],[0;0],[0.4;04]);
    xopt (i) = inputs(1);
    yopt (i) = inputs(2);
end

study2_time = toc;

%plotting2
figure(2)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','O','markersize',3,'markerfacecolor','g')
title('Bounded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('function minimum')

%increrasing ga iteration 

options = optimoptions('ga');
options = optimoptions(options,'populationsize',1000);

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;

%plotting2
figure(3)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','O','markersize',3,'markerfacecolor','g')
title('Bounded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('function minimum')

for this there is 3 studies

For global maxima

figure1

For finding minimum function by ga

Study1

figure1

Study2

figure2

Study3

figure3

Result:

• The code for global maxima and stalagmite function is done.
• The concept of genetic algoritham and the syntax is done.
• The graph for all three studies and global maxima are plotted.