Week 5 - Genetic Algorithm

Aim:- To optimise the stalagmite function and find the global maxima of the function using Genetic algorithm

Equations:- Since for this there is no Equation , but we have the a syntax for this i.e..

f=ga(fun,nvars,A,b,[],[],lb,ub,nonlcon,IntCon,options)

fun - fitness function.

nvars- number of variables,

A,b= Linear inequality constraints

Aeq,beq=Linear equality constraints

lb- Lower bound(limit),

ub- Upper bound(limit),

nonlcon : Use for nonlinear constraints

IntCon : to get solution in integer.

options- To set population size minimize with non-default options

Objective of project:-

The objective of project is to get the best possible solution for a given problem.

Body of content:-

% A program for a GA

clear all
close all
clc

% defining space
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
 
num_case = 50 ;

% creating a 2dimensional 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

% case 1:- statistical behaviour
tic
for i =1:num_case
 [inputs,fopt(i)]= ga(@stalagmite,2);
 xopt(i)=inputs(1);
 yopt(i)= inputs(2);
end
study1_time=toc;

% plotting
figure(1)
subplot(2,1,1)
hold on
surfc(xx,yy,-F)
shading interp 
plot3(xopt,yopt,fopt,'marker','o','MarkerFaceColor','r');
title('unbounded Inputs')
subplot(2,1,2)
plot(fopt)
xlabel('interations')
ylabel('Function minimum')

%case 2:- statistical behaviour with upper and lower bounds
tic
for i =1:num_case

 [inputs,fopt(i)]= ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]);
 xopt(i)=inputs(1);
 yopt(i)= inputs(2);
end
study2_time=toc;

% plotting
figure(2)
subplot(2,1,1)
hold on
surfc(xx,yy,-F)
shading interp 
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfaceColor','r')
title('bounded Inputs')
subplot(2,1,2)
plot(fopt)
xlabel('interations')
ylabel('Function minimum')

%case 3:- Increasing ga interations
tic
options=optimoptions('ga');
options= optimoptions(options,'PopulationSize',170);

for i =1:num_case

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

% plotting
figure(3)
subplot(2,1,1)
hold on
surfc(xx,yy,-F)
shading interp 
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfaceColor','r')
title('population size')
subplot(2,1,2)
plot(fopt)
xlabel('interations')
ylabel('Function minimum')

function:-
function F=stalagmite(input_vector)
% input_vector(1)=x
% input_vector(2)=y


term1= (sin(5.1*pi*input_vector(1)+0.5))^6;
term2= exp(-4*log(2)*((input_vector(1)-0.0667)^2/0.64));
term3= (sin(5.1*pi*input_vector(2)+0.5))^6 ;
term4= exp(-4*log(2)*((input_vector(2)-0.0667)^2/0.64));

F= -(term1*term2*term3*term4);
end

Here we can say that A program of Genetic algorithm is ready to go with a proper execution, well in these we have used the commands like clc ,clear all ,Inputs , plot , x-label ,y-label.

Commands use for: -

• Clear all = used to clear the workspace.
• CLC = used to clear the command window.
• Close all = used to close the background program.
• % Inputs = used this command for better clarification to enter or add new terms.
• Plot = used for plotting a graph, terms used in bracket as shown above , always the same terms means exact words or letters used in Inputs so that we won’t get errors will running program.
• x-label & y-label = used for name recognition in x-axis & y-axis
• Hold on = Adding new plots or frame.
• Hold off = Hold the new plots or frame.
• For loop = To repeat the specified number of times.
• Axis = We used this command to inbuilt the plot or frame inside it , we can say like a frame of this dimensions.
• Legend = used this command to show a label in a plot itself
• Title = used this command to give heading .
• linewidth = used this command to increase the thickness of line.
• Tic = start stopwatch timer.
• Toc = The actual time taken to complete.
• Shading = The shading controls the colour shading of surface and patch graphics objects.
• Surfc = Contour plot under surface plot

Explanation of GA & syntax:-

>Genetic algorithm is used to find out the best possibilities of solution for the given problem , A genetic algorithm is nothing but a method to try and try to get the best result for a particular problem , In these GA there are two optimization i.e.. constraint and unconstraint optimization. The constraint optimization has a bounded conditions to get the stable result and for the unconstraint optimization has a unbounded conditions to get the unstable result.

f=ga(fun,nvars,A,b,[],[],lb,ub,nonlcon,IntCon,options)

where,

fun = fitness function.

nvars = number of variables.

A,b = Linear inequality constraints.

Aeq,beq = Linear equality constraints.

lb = Lower bound(limit).

ub = Upper bound(limit).

nonlcon = Use for nonlinear constraints.

IntCon = to get solution in integer.

options = To set population size minimize with non-default options.

Stopping criteria of GA:-

As soon the algorithm reaches to the point then it will get stop.