Week 4.1 - Genetic Algorithm

AIM
To study about Genetic Algorithm and to calculate the global maxima of stalagmite function using Genetic Algorithm in MATLAB.

OBJECTIVE
To study about Genetic Algorithm and stalagmite function.
To write a code in Matlab to optimise the stalagmite function and find the global maxima of the function.
To plot graphs based on the studies done for genetic Algorithm.

Matlab code

%Genetic algorithm
clear all
close all
clc

%independent variacles
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
%matrix
[xx,yy]=meshgrid(x,y);
%Number of iterations
num_cases=50;
% starting the loop
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
% creates a 3d surface
figure(1)
surfc(xx,yy,-f)
shading interp
xlabel('x value')
ylabel('y value')
title('stalagmite function')

%study 1
tic
for i=1:num_cases
[inputs,fopt(i)]=ga(@stalagmite,2);
xopt(i)= inputs(1);
yopt(i)= inputs(2);
end
study_1=toc
figure(2)
subplot(2,1,1)
hold on
surfc(x,y,-f)
title('Unbounded inputs')
xlabel('x value')
ylabel('y value')

shading interp
hold on

plot3(xopt,yopt,-fopt,'marker','o','MarkerSize',5,'MarkerFaceColor','r')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('function maximum')

%case 2nd with upper and lower bounds

tic;
for i=1:num_cases
[inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[0.6;0.6]);
xopt(i)=inputs(1);
yopt(i)=inputs(2);
end
study_2_time=toc
figure(3)
subplot(2,1,1)
surfc(x,y,-f)
shading interp
title('Bounded inputs')
xlabel('x value')
ylabel('y value')

hold on

plot3(xopt,yopt,-fopt,'marker','o','MarkerSize',5,'MarkerFaceColor','r')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('function maximum')

%study 3 increasing the population size
% by doing so we are increasing the probablity of the function
% to converge at global maximum
options=optimoptions(@ga)
options=optimoptions(@ga,'populationSize',500)

tic
for i=1:num_cases
[inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[0.6;0.6],[],[],options);
xopt(i)=inputs(1);
yopt(i)=inputs(2);
end
study_3_time=toc
figure(4)
subplot(2,1,1)
surfc(x,y,-f)
shading interp
title('bounded inputs with increased population size')
xlabel('x value')
ylabel('y value')

hold on

plot3(xopt,yopt,-fopt,'marker','o','MarkerSize',4,'MarkerFaceColor','r')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('function maximum')

function [f] = stalagmite(input_vector)

a=input_vector(1);
b=input_vector(2);

term1=(sin((5.1*pi*a)+0.5))^6;
term2=exp(-4*log(2)*((a-0.0667)^2)/0.64);
term3=(sin((5.1*pi*b)+0.05))^6;
term4=exp((-4)*log(2)*((b-0.0667)^2)/0.64);

[f]=-(term1*term2*term3*term4);
end

CODE EXPAINATION
1. The code starts with clear all, close all and clc where,
‘clear all’ command is used to clear the data stored in the workspace.
‘close all’ command is used to close all plots and figures.
‘clc’ command is used to clear the command window.
2. Then the ‘x’ array and ‘y’ array is defined using ‘linspace’ command to create values at 150 equal intervals.
3. A 2D mesh of x and y array is created by using ‘meshgrid’ command and the number of iterations or cases that the code should perform is defined. This is given by n = 50.
4. To evaluate the stalagmite function, ‘for’ loop is used where ‘i’ is initialized and for every value of ‘i’ another ‘for’ loop is used to initialize ‘j’.
5. The input vector values of ‘xx’ and ‘yy’ are defined and the stalagmite function for these input vector is defined. This will form a basic stalagmite function and we can plot it separately if needed.
6. After preparing the inputs needed for the function, next step is to obtain the results for 3 cases of the genetic algorithm. In first case or study, the maximum value of function is found out with no restrictions, no bounds and no limits.
7. This is done using ‘for’ loop where we use ‘ga’ syntax to give values for inputs and optimum max value. The stalagmite function is called in ‘ga’ syntax and number of variables ‘nvars’ are 2.
8. To know the time that is taken to solve the loop for 50 iterations, ‘tic’ and ‘toc’ command is used before and after the loop respectively.
9. Then the function created in the loop is plotted by using ‘surfc’ command for ‘xx, yy and function output’. To neglect the gridlines on the surface and obtain a smooth and clean plot, ‘shading interp’ command is used. The ‘subplot(2,1,1)’ is used to plot multiple plots on same figure which also divides figure in grid of 2x1 and axes to be placed at 1.
10. The ‘plot3’ command plots the 3D points at ‘xopt, yopt and optimum value of function’ with ‘o’ shaped and red colour. The labels and title are then provided. Now in the same figure, the values of optimum function ‘fopt’ are plotted for 50 iterations.
11. The results of the above case will be different for each time we run the program. To get same value each time, we need to restrict the program within specific bounds namely upper bound and lower bound. The procedure for coding and plottind this procedure is same as Study 1. The only difference is in the ‘ga’ syntax. This time the algorithm is provided lower limits and upper limits as [0;0] and [1;1] respectively.
12. After plotting the results for Study 2 or the bounded case, it is observed that the results are not same every time the program terminates. A solution to this problem is to increase the number of cases or iterations.
13. So for Study 3, the number of iterations are increased to 300. So for this study, the syntax, “options = optimoptions(‘SolverName’,’Name’,‘Value’)” is used which returns with named parameter ‘Populations’ with the altered values ‘300’.
14. The syntax for ‘ga’ contains ‘option’ term and the rest of the code is same as for Study 1 and 2.

RESULTS

CONCLUSION

In this challenge , the concept of Genetic algorithm was used to optimize the stalagmite function to find the global maxima. The concept and working of genetic algorithm are studied and used in the three cases to optimize the function and find the global maxima.