Challenge

Week 5 - Genetic Algorithm

HHarikrishnan B17 Oct 2022

MATLAB

AIM:

To write a program in Matlab to optimise the stalagmite function and find the global maxima of the function.

SOFTWARE USED:

Matlab

GOVERNING EQUATIONS:

OBJECTIVES:

Program code in MATLAB to optimise the stalagmite function and find the global maxima of the function.

Explanation of the concept of genetic algorithm and also explain the syntax for gain MATLAB in the report.
Plot graphs for all 3 studies and for F maximum vs no. of iterations with title and axis labelled.

THEORY:

GENETIC ALGORITHM

The genetic Algorithm is a non traditional optimization method which is stochastic in nature.The genetic Algorithm is based on natural evolution. It includes the Initiation of an initial population and the selection of best-fit input data for crossover and mutation to get an optimized maximum output for the function for which the genetic algorithm is used. This process continues for a some random population for number of iterations till the best solutions are obtained. Normally GA are used for maximization problems, but it can be also used for minimization problems.

STALAGMITE EQUATION

Stalagamites are certain rocks which are found in caves which develops due to accumulation of calcium deposites through the rock gaps. These rocks will form a typical shape which is being used as a function in mathematics to find out maximum value of a function. In this problem, Stalagmite function will be optimized using genetic algorithm approach. It helps in generating local minimas and maximas.

BODY OF CONTEXT:

PROGRAM CODE:

% Program to optimise the stalagmite function and find the global maxima of the function.

clear all; % clear all variables in workspace

close all;% close all plots

clc;%clear command window

%Inputs

num_cases=150;

%Initializing x and y inputs

x=linspace(0,0.6,150);

y=linspace(0,0.6,150);

[xx yy]=meshgrid(x,y);%Creating a meshgrid for all points and assigning each values to xx and yy

%Generating data for plot

for i=1:length(xx)

for j=1:length(yy)

input_vector(1)=xx(i,j); %Storing each value of xx to row side of input_vector

input_vector(2)=yy(i,j); %Storing each value of yy to column side of input_vector

f(i,j)= stalagmite(input_vector); %Calling stalagmite function

end

surfc(xx,yy,f) %Plotting surface

xlabel('x values'); % giving x label as x values

ylabel('y values'); % giving y label as y values

shading interp %shading interp to avoid grid lines on graph

%1st case

tic %Initiating tic for time calculation starting

for i=1:num_cases % For loop for settinmg number of iterations as 150

[inputs,fopt(i)]= ga(@stalagmite,2); % Calling ga function with 2 inputs

xopt(i)=inputs(1); %Assigning row values of input to xopt

yopt(i)=inputs(2); %Assigning column values of input to xopt

end

fopt=-fopt; %Since minimization value of fopt will be negative, so to make it positive

time_study1=toc % Initiating toc for time calculation ending

%Creating plot

figure(1) %Creating figure 1

subplot(2,1,1) %Creating subplot with 2 rows and in first row

hold on %Setting hold on

surfc(x,y,-f) %Plotting x,y and -f

xlabel('x values'); %Setting x label

ylabel('y values'); %Setting y label

shading interp %shading interp to avoid grid lines on graph

plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfacecolor','r') %Creating 3d plot of xopt,yopt and fopt and

% giving marker of 5pts with red colour for datapoints

title('Unbounded inputs') %Setting title

%figure(2)

subplot(2,1,2) %Creating subplot with 2 rows and in second row

plot(fopt) % plot fopt

xlabel('Iteration') %giving x label

ylabel('Function minimum') %giving y label

% 2nd Case

tic %Initiating tic for time calculation starting

for i=1:num_cases % For loop for settinmg number of iterations as 150

[inputs2,fopt2(i)]= ga(@stalagmite,2,[],[],[],[],[0,0],[1,1]); % Calling ga function with 2 inputs

%and setting A,B,Aeq,Beq to null vectors showing no inequalies and setting

%lowerbound as 0,0 and upper bound as 1,1

% Calling ga function with 2 inputs

xopt2(i)=inputs2(1); %Assigning row values of input to xopt2

yopt2(i)=inputs2(2); %Assigning column values of input to xopt2

end

fopt2=-fopt2; %Since minimization value of fopt will be negative, so to make it positive

time_study2=toc% Initiating toc for time calculation ending

%Creating plot

figure(2) %Creating figure 2

subplot(2,1,1) %Creating subplot with 2 rows and in first row

hold on %Setting hold on

surfc(x,y,-f) %Plotting x,y and -f

xlabel('x values'); %Setting x label

ylabel('y values'); %Setting y label

shading interp %shading interp to avoid grid lines on graph

plot3(xopt2,yopt2,fopt2,'marker','o','markersize',5,'markerfacecolor','r')%Creating 3d plot of xopt2,yopt2and fopt2 and

% giving marker of 5pts with red colour for datapoints

title('Bounded inputs') %Setting title

%figure(2)

subplot(2,1,2) %Creating subplot with 2 rows and in second row

plot(fopt2) % plot fopt2

xlabel('Iteration') %Setting x label

ylabel('Function minimum') %Setting y label

%3rd case: Maximising the population size to optimise the result

options= optimoptions ('ga'); %Setting options using optimoptions

options=optimoptions(options,'PopulationSize',500); %Setting population size to 500 as option

tic %Initiating tic for time calculation starting

for i=1:num_cases % For loop for settinmg number of iterations as 150

[inputs3, fopt3(i)]=ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6],[],[],options);% Calling ga function with 2 inputs

%and setting A,B,Aeq,Beq to null vectors showing no inequalies and setting

%lowerbound as 0,0 and upper bound as 0.6,0.6

%Seeting c and d inequalities as null vector and caaling options

xopt3(i)=inputs3(1); %Assigning row values of input3 to xopt3

yopt3(i)=inputs3(2); %Assigning column values of input3 to xopt3

end

figure(3) %Creating figure 3

subplot(2,1,1) %Creating subplot with 2 rows and in first row

hold on %Setting hold on

surfc(x,y,-f) %Plotting x,y and -f

xlabel('x values'); %Setting x label

ylabel('y values'); %Setting y label

shading interp %shading interp to avoid grid lines on graph

plot3(xopt3,yopt3,-fopt3,'marker','o','markersize',5,'markerfacecolor','r')%Creating 3d plot of xopt3,yopt3and fopt3 and

% giving marker of 5pts with red colour for datapoints

title('Bounded inputs with optimoptions function')%Setting title

subplot(2,1,2) %Creating subplot with 2 rows and in second row

plot(-fopt3) % plot fopt2

xlabel('Iteration') %Setting xlabel

ylabel('Function minimum') %setting y label

time_study3=toc

% Initiating toc for time calculation ending

% Program to optimise the stalagmite function and find the global maxima of the function.
clear all; % clear all variables in workspace 
close all;% close all plots 
clc;%clear command window
%Inputs
num_cases=150;
%Initializing x and y inputs
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
[xx yy]=meshgrid(x,y);%Creating a meshgrid for all points and assigning each values to xx and yy


%Generating data for plot
for i=1:length(xx)
  for j=1:length(yy)
    input_vector(1)=xx(i,j); %Storing each value of xx to row side of input_vector
    input_vector(2)=yy(i,j); %Storing each value of yy to column side of input_vector
    f(i,j)= stalagmite(input_vector); %Calling stalagmite function
  end
end
surfc(xx,yy,f) %Plotting surface
xlabel('x values'); % giving x label as x values
ylabel('y values'); % giving y label as y values
shading interp %shading interp to avoid grid lines on graph
%1st case
tic %Initiating tic for time calculation starting
for i=1:num_cases % For loop for settinmg number of iterations as 150
 [inputs,fopt(i)]= ga(@stalagmite,2); % Calling ga function with 2 inputs
  xopt(i)=inputs(1); %Assigning row values of input to xopt
  yopt(i)=inputs(2); %Assigning column values of input to xopt
end
fopt=-fopt; %Since minimization value of fopt will be negative, so to make it positive
time_study1=toc % Initiating toc for time calculation ending
%Creating plot
figure(1) %Creating figure 1
subplot(2,1,1) %Creating subplot with 2 rows and in first row
hold on %Setting hold on
surfc(x,y,-f) %Plotting x,y and -f
xlabel('x values'); %Setting x label
ylabel('y values'); %Setting y label
shading interp %shading interp to avoid grid lines on graph
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfacecolor','r') %Creating 3d plot of xopt,yopt and fopt and 
% giving marker of 5pts with red colour for datapoints 
title('Unbounded inputs') %Setting title
%figure(2)
subplot(2,1,2) %Creating subplot with 2 rows and in second row
plot(fopt) % plot fopt
xlabel('Iteration') %giving x label
ylabel('Function minimum') %giving y label

% 2nd Case
tic %Initiating tic for time calculation starting
for i=1:num_cases % For loop for settinmg number of iterations as 150
 [inputs2,fopt2(i)]= ga(@stalagmite,2,[],[],[],[],[0,0],[1,1]); % Calling ga function with 2 inputs
 %and setting A,B,Aeq,Beq to null vectors showing no inequalies and setting
 %lowerbound as 0,0 and upper bound as 1,1
 % Calling ga function with 2 inputs
  xopt2(i)=inputs2(1); %Assigning row values of input to xopt2
  yopt2(i)=inputs2(2); %Assigning column values of input to xopt2
end
fopt2=-fopt2; %Since minimization value of fopt will be negative, so to make it positive
time_study2=toc% Initiating toc for time calculation ending
%Creating plot
figure(2) %Creating figure 2
subplot(2,1,1) %Creating subplot with 2 rows and in first row
hold on %Setting hold on
surfc(x,y,-f) %Plotting x,y and -f
xlabel('x values'); %Setting x label
ylabel('y values'); %Setting y label
shading interp %shading interp to avoid grid lines on graph
plot3(xopt2,yopt2,fopt2,'marker','o','markersize',5,'markerfacecolor','r')%Creating 3d plot of xopt2,yopt2and fopt2 and 
% giving marker of 5pts with red colour for datapoints 
title('Bounded inputs') %Setting title
%figure(2)
subplot(2,1,2) %Creating subplot with 2 rows and in second row
plot(fopt2) % plot fopt2
xlabel('Iteration') %Setting x label
ylabel('Function minimum') %Setting y label

%3rd case: Maximising the population size to optimise the result
options= optimoptions ('ga'); %Setting options using optimoptions
options=optimoptions(options,'PopulationSize',500); %Setting population size to 500 as option

tic %Initiating tic for time calculation starting
for i=1:num_cases % For loop for settinmg number of iterations as 150
  [inputs3, fopt3(i)]=ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6],[],[],options);% Calling ga function with 2 inputs
 %and setting A,B,Aeq,Beq to null vectors showing no inequalies and setting
 %lowerbound as 0,0 and upper bound as 0.6,0.6
 %Seeting c and d inequalities as null vector and caaling options 
  xopt3(i)=inputs3(1); %Assigning row values of input3 to xopt3
  yopt3(i)=inputs3(2); %Assigning column values of input3 to xopt3
end
figure(3) %Creating figure 3
subplot(2,1,1) %Creating subplot with 2 rows and in first row
hold on %Setting hold on
surfc(x,y,-f) %Plotting x,y and -f
xlabel('x values'); %Setting x label
ylabel('y values'); %Setting y label
shading interp %shading interp to avoid grid lines on graph
plot3(xopt3,yopt3,-fopt3,'marker','o','markersize',5,'markerfacecolor','r')%Creating 3d plot of xopt3,yopt3and fopt3 and 
% giving marker of 5pts with red colour for datapoints 
title('Bounded inputs with optimoptions function')%Setting title
subplot(2,1,2) %Creating subplot with 2 rows and in second row
plot(-fopt3) % plot fopt2
xlabel('Iteration') %Setting xlabel
ylabel('Function minimum') %setting y label
time_study3=toc
% Initiating toc for time calculation ending

FUNCTION CODE:

%Function to perform and calculate stalagmite

function[f]= stalagmite(input_vector)

fx=(sin(5.1*pi*input_vector(1)+0.5))^6; %Initiating equation 1

fy=(sin(5.1*pi*input_vector(2)+0.5))^6; %Initiating equation 2

fxx=exp((-4*log(2)*(((input_vector(1)-0.0667)^2)/0.64))); %Initiating equation 3

fyy=exp(-4*log(2)*(((input_vector(2)-0.0667)^2)/0.64)); %Initiating equation 4

f=-(fx*fy*fxx*fyy); %Initiating final equation,'-' sign makes stalagmite function to appear in opposite direction in plot to get maxima(not minima)

end

%Function to perform and calculate stalagmite 
function[f]= stalagmite(input_vector)
  fx=(sin(5.1*pi*input_vector(1)+0.5))^6; %Initiating equation 1
  fy=(sin(5.1*pi*input_vector(2)+0.5))^6; %Initiating equation 2

  fxx=exp((-4*log(2)*(((input_vector(1)-0.0667)^2)/0.64))); %Initiating equation 3
  fyy=exp(-4*log(2)*(((input_vector(2)-0.0667)^2)/0.64)); %Initiating equation 4
  
  f=-(fx*fy*fxx*fyy); %Initiating final equation,'-' sign makes stalagmite function to appear in opposite direction in plot to get maxima(not minima)
end

SYNTAX FOR PROGRAM WITH EXPLANATION:

<--Inputs

num_cases=150;

<--Initializing x and y inputs

x=linspace(0,0.6,150);

y=linspace(0,0.6,150);

[xx yy]=meshgrid(x,y);<--Creating a meshgrid for all points and assigning each values to xx and yy

<--Generating data for plot

for i=1:length(xx)

for j=1:length(yy)

input_vector(1)=xx(i,j); <--Storing each value of xx to row side of input_vector

input_vector(2)=yy(i,j); <--Storing each value of yy to column side of input_vector

f(i,j)= stalagmite(input_vector); <--Calling stalagmite function

end

surfc(xx,yy,f) <--Plotting surface

xlabel('x values'); <-- giving x label as x values

ylabel('y values'); <-- giving y label as y values

shading interp <--shading interp to avoid grid lines on graph

<--1st case

tic <--Initiating tic for time calculation starting

for i=1:num_cases <-- For loop for settinmg number of iterations as 150

[inputs,fopt(i)]= ga(@stalagmite,2); <-- Calling ga function with 2 inputs

xopt(i)=inputs(1); <--Assigning row values of input to xopt

yopt(i)=inputs(2); <--Assigning column values of input to xopt

end

fopt=-fopt; <--Since minimization value of fopt will be negative, so to make it positive

time_study1=toc <-- Initiating toc for time calculation ending

<--Creating plot

figure(1) <--Creating figure 1

subplot(2,1,1) <--Creating subplot with 2 rows and in first row

hold on <--Setting hold on

surfc(x,y,-f) <--Plotting x,y and -f

xlabel('x values'); <--Setting x label

ylabel('y values'); <--Setting y label

shading interp <--shading interp to avoid grid lines on graph

plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfacecolor','r') <--Creating 3d plot of xopt,yopt and fopt and

<-- giving marker of 5pts with red colour for datapoints

title('Unbounded inputs') <--Setting title

<--figure(2)

subplot(2,1,2) <--Creating subplot with 2 rows and in second row

plot(fopt) <-- plot fopt

xlabel('Iteration') <--giving x label

ylabel('Function minimum') <--giving y label

<-- 2nd Case

tic <--Initiating tic for time calculation starting

for i=1:num_cases <-- For loop for settinmg number of iterations as 150

[inputs2,fopt2(i)]= ga(@stalagmite,2,[],[],[],[],[0,0],[1,1]); <-- Calling ga function with 2 inputs

<--and setting A,B,Aeq,Beq to null vectors showing no inequalies and setting

<--lowerbound as 0,0 and upper bound as 1,1

<-- Calling ga function with 2 inputs

xopt2(i)=inputs2(1); <--Assigning row values of input to xopt2

yopt2(i)=inputs2(2); <--Assigning column values of input to xopt2

end

fopt2=-fopt2; <--Since minimization value of fopt will be negative, so to make it positive

time_study2=toc<-- Initiating toc for time calculation ending

<--Creating plot

figure(2) <--Creating figure 2

subplot(2,1,1) <--Creating subplot with 2 rows and in first row

hold on <--Setting hold on

surfc(x,y,-f) <--Plotting x,y and -f

xlabel('x values'); <--Setting x label

ylabel('y values'); <--Setting y label

shading interp <--shading interp to avoid grid lines on graph

plot3(xopt2,yopt2,fopt2,'marker','o','markersize',5,'markerfacecolor','r')<--Creating 3d plot of xopt2,yopt2and fopt2 and

<-- giving marker of 5pts with red colour for datapoints

title('Bounded inputs') <--Setting title

<--figure(2)

subplot(2,1,2) <--Creating subplot with 2 rows and in second row

plot(fopt2) <-- plot fopt2

xlabel('Iteration') <--Setting x label

ylabel('Function minimum') <--Setting y label

<--3rd case: Maximising the population size to optimise the result

options= optimoptions ('ga'); <--Setting options using optimoptions

options=optimoptions(options,'PopulationSize',500); <--Setting population size to 500 as option

tic <--Initiating tic for time calculation starting

for i=1:num_cases <-- For loop for settinmg number of iterations as 150

[inputs3, fopt3(i)]=ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6],[],[],options);<-- Calling ga function with 2 inputs

<--and setting A,B,Aeq,Beq to null vectors showing no inequalies and setting

<--lowerbound as 0,0 and upper bound as 0.6,0.6

<--Setting c and d inequalities as null vector and calling options

xopt3(i)=inputs3(1); <--Assigning row values of input3 to xopt3

yopt3(i)=inputs3(2); <--Assigning column values of input3 to xopt3

end

figure(3) <--Creating figure 3

subplot(2,1,1) <--Creating subplot with 2 rows and in first row

hold on <--Setting hold on

surfc(x,y,-f) <--Plotting x,y and -f

xlabel('x values'); <--Setting x label

ylabel('y values'); <--Setting y label

shading interp <--shading interp to avoid grid lines on graph

plot3(xopt3,yopt3,-fopt3,'marker','o','markersize',5,'markerfacecolor','r')<--Creating 3d plot of xopt3,yopt3and fopt3 and

<-- giving marker of 5pts with red colour for datapoints

title('Bounded inputs with optimoptions function')<--Setting title

subplot(2,1,2) <--Creating subplot with 2 rows and in second row

plot(-fopt3) <-- plot fopt2

xlabel('Iteration') <--Setting xlabel

ylabel('Function minimum') <--setting y label

time_study3=toc

<-- Initiating toc for time calculation ending

SYNTAX FOR FUNCTION WITH EXPLANATION:

<--Function to perform and calculate stalagmite

function[f]= stalagmite(input_vector)

fx=(sin(5.1*pi*input_vector(1)+0.5))^6; <--Initiating equation 1

fy=(sin(5.1*pi*input_vector(2)+0.5))^6; <--Initiating equation 2

fxx=exp((-4*log(2)*(((input_vector(1)-0.0667)^2)/0.64))); <--Initiating equation 3

fyy=exp(-4*log(2)*(((input_vector(2)-0.0667)^2)/0.64)); <--Initiating equation 4

f=-(fx*fy*fxx*fyy); <--Initiating final equation,'-' sign makes stalagmite function to appear in opposite direction in plot to get maxima(not minima)

end

CONCLUSION: