Week 4.1 - Genetic Algorithm

It defines a set of lower and upper bounds on the design variables, x, so that a solution is found in the range lb ≤ x ≤ ub. (Set Aeq=[] and beq=[] if no linear equalities exist.)

%DEFINING THE FUNCTION

function [F] = stalagmite(inputs)

% GIVING THE INPUT VARIABLES

x=inputs(1);
y=inputs(2);

% BREAKING THE MAIN FUNCTION CODE 

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

% FINAL FUNCTION

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

In this we have written a code to define stalagmite function.Here we broke down the formula into 4 terms for ease convenience.

clear all
close all
clc

%Defining our search space

x=linspace(0,1,100);
y=linspace(0,1,100);

%Creating 2d mesh

[xx yy]= meshgrid(x,y);

% Evaluating the stalagmite function

for i=1:length(xx)
    for j=1:length(yy)
        
        inputs(1)=xx(i,j);
        inputs(2)=yy(i,j);
        F(i,j)= stalagmite(inputs);
    end
end

num=100;
tic
%Case 1 Stastical behaviour

for i=1:num
    [inputs,Fout(i)]= ga(@stalagmite,2);
    xout(i)=inputs(1);
    yout(i)=inputs(2);
end
% PLOTTING THE GRAPH

case1_time=toc;
figure(1)
subplot(2,1,1)
surfc(xx,yy,-F)
shading interp
hold on
plot3(xout,yout,-Fout,'marker','*','markersize',5,'markerfacecolor','r')
title('Unbounded Inputs')
subplot(2,1,2)
plot(-Fout)
xlabel('Iterations')
ylabel('Function Maximum')
tic

%CASE 2 STASTICAL BEHAVIOUR WITH UPPER & LOWER BOUND

for i=1:num
    
    [inputs,Fout(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[0.3;0.3],[],[]);
    xout(i)=inputs(1);
    yout(i)=inputs(2);
end

% PLOTTING THE GRAPH

figure(2)
subplot(2,1,1)
surfc(x,y,-F)
shading interp
hold on
plot3(xout,yout,-Fout,'marker','*','markersize',5,'markerfacecolor','g')
title('Bounded Inputs')
subplot(2,1,2)
plot(-Fout)
xlabel('Iterations')
ylabel('Function Maximum')

% Case 3 Increasing Iterations

options=optimoptions('ga');
options=optimoptions(options,'populationsize',200);

for i= 1:num
    [inputs,Fout(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);
    xout(i)=inputs(1);
    yout(i)=inputs(2);
end

%PLOTTING THE GRAPH

figure(3)
subplot(2,1,1)
surfc(x,y,-F)
shading interp
hold on
plot3(xout,yout,-Fout,'marker','*','markersize',5,'markerfacecolor','b')
title('Bounded Inputs with Increased Population')
subplot(2,1,2)
plot(-Fout)
xlabel('Iterations')
ylabel('Function maximum')

STEPS TAKEN IN PROGRAMMING:

• Firstly we write the code to define the stalagmite function where we broke it into 4 terms for easy calculation.
• After that we write the main code where we define two inputs x and y by naming it as our search space bith of which are one dimensional arrays containing 100 elements each.
• Now by using the meshgrid command we created a 2d arrays for x and y.
• Now by using the for loop we will enter the 2d array valuews into the stalagmite function. Here we encountered a problem that everytime we executed the code it gives us the new results so to overcome that problem we introduced a new for loop. We then input hese values in genetic algorithm where (ga) function is used.
• After this we need to improve our genetic algorithm, for that we took three cases to optimise the function so that we can exactly find global maxima for given function.
• In case 1 we evaluate all possible global maxima for all the iterations using a loop. we made two plots in one figure using subplot command. 'surfc' command is used for x and y graph and for function max vs itertions we used simple plot command.
• In casse 2 we put an upper and lower bound on ga function and agian plotted two plots.
• In case 3 we have increased the number of iterations because in GA successive iterationss, the population evolves toward an optimal solution i.e more the iterations more optimal solution will be and the same commands for plotting were used.

RESULTS:

CASE 1-

We can see here that we have solutions all over the graph so due to that we can not find our optimal solution easily.

CASE 2-

In this case we can see that the values are in betwwn 0.85 to 1 which is not a much difference so this is better than the previous case.

CASE 3-

ERRORS :

The main error which i faced was in the graph that it was not proper so to overcome that error i tried to putting (-) sign in all F values and in main function F.

same erroer happened in all the three cases and was resolved.

REFERENCES:

https://in.mathworks.com/help/gads/ga.html

https://towardsdatascience.com/introduction-to-genetic-algorithms-including-example-code-e396e98d8bf3