Week 4.1 - Genetic Algorithm

 

Matlab Program to optimise the stalagmite function and finding the global maxima

 

1.Basics Concepts:

 

1.1 optimization:

Optimization is the area of study which is used to maximize or minimize the particular function, which depends upon multiple parameters to get the desired the output.

1.2 Stalagmite Mathematical model:

Stalagmite Mathematical model used to develop the local maxima or minima . Here to determine the optimum solution of the maxima of the function by using the Genetic algorithm technique.

1.3 Genetic algorithm:

Genetic algorithm is an optimization technique use to slove Non-linear and Non-differential optimization problems.The name derived from Evolution Biology Technique(Darwin’s Theory of Evolution)

It starts with a initial generation of a candidate solution that is tested against the objective function. Subsequent Generation evolves from the first through selection crossover and mutation, which continues until the cycle stop with true value

Initialization

 

Selection

Crossover

 

Mutation

Let, take a exampleSelection:In this we are selecting best Performing bit strings from one generation to another generation .i.e based on the fitness

Here we have two parents who have good performance so selecting them based on fitness

1.parent1 =[1 0 1 0 0 1 1 0 0 0]

2.parent2 =[1 0 0 1 0 0 1 0 1 0]

Crossover:Due to they performed well ,therefore taking between them to create a child as shown below

1.parent1 =[1 0 1 0 0 1 1 0 0 0]

2.parent2 =[1 0 0 1 0 0 1 0 1 0]

3.child =[1 0 0 0 0 1 1 0 1 0]

Mutation:Taking parent and mutate certain variable and create a child based on mutation I.e in mutation we are adding other qualities to thechild

2.Procedure:

To solve the global optimization problem we are using the stalagmite Mathematical model which is used to develop the local maxima and minima ,but in our problem we have to get global maxima of the function hence we are using genetic algorithm

2.1 Initialization

First we are defining our search space and creating the one dimensional array Xand Y with linspace command then Xand y are the inputs array does not define the workspace for thegenetic algorithm as follows

x=linspace(0,0.6,150):

y=linspace(0,0.6,150):

Where:

0 is an initial value

0.6 is an final value at last

150 is number of terms

Now creating the two-dimensional array using meshgrid as follows:

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

for each value of x and y ,we are going to create a input vector (row vector) I.e has two coloumns

Note: the command line [xx,yy]=meshgrid(x,y) is to beused in seperate function name as “stalagmite_function” which explanation we can get in error and elimination section further.

Further we are using the for loop to getthe same action multiple times

for i=1:length(x)

for j=1:length(y)

input_vector(1)=xx(i,j);

input_vector(2)=yy(i,j);

f(i,j)=stalagmite_function(input_vector);

end

end

where the stalagmite is a seperate function which is based on the formula as follows:

1_x=(sin(5.1*pi*x+0.5)).^6;

f2_y=(sin(5.1*pi*y+0.5)).^6;

f3_x=exp((-4*0.6931).*(x-0.0667)^2/(0.64));

f4_y=exp((-4*0.6931).*(x-0.0667)^2/(0.64));

stalagmite=(f1_x.*f2_y.*f3_x.*f4_y)

in main program we are going to use the one dimensional arrays xand y with linspace command and we know that x and y are input array it does not define the workspace for the workspace for the genetic algorithm

 

Program for different optimization studies:

In this program we are going to do three study

Initially ,we use to one dimensional array x and y with linspace command as we know that x and y are the input array and does not define the workspace of the genetic algorithm ,Here weare using the component used in the program as well

Study_1:

In this ,we are calling the ga function and passing a number of variables as an input which is equal to 2 and strong x andy value also the (fopt) the optimum value which the function is produced with out boundary condition

After opening the new figure window [figure1] and creating the subplot (which creates the multiple plots in the same figure window)The first plot is surf command of optimum value predicts over a 50 studiesThe second plot is which directly optimum value is get plotted as a function of iteratations

Study_2:

In study 2 we are going to use the upper and lowerlimits which is known as upper and lower bounds The syntax use is based on our reffered syntax section

Study_3:

To optimize the result and getting more accuracy we are increasing the iteration in genetic algorithm. Because by default the ga takes 50value .here we are going to increase the population size because if initial population is too small then it is not going to find out global maxima or minima .it only finds the local minima ans maxima

 

 

% To find global maxima forastalagmote function

clear all
close all
clc
% input
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
num_cases=50

% 2d forming array
[xx,yy]=meshgrid(x,y);
% Evaluating the stalagmite_function
for i=1:length(x)
    for j=1:length(y)
       input_vector(1)=xx(i,j);
       input_vector(2)=yy(i,j);
       f(i,j)=stalagmite_function(input_vector);
     end
   end


tic
% study 1- stalagmite behaviour with out any specified boundries
for i =1:num_cases
[inputs,fopt1(i)]=ga(stalagmite_function,2);
xopt1(i)=inputs(1);
yopt1(j)=inputs(2);
end
study1_time=toc

figure(1)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt1,yopt1,fopt1,'Marker','o','markersize',5,'markerfacecolor','r');
title('unbounded inputs')
subplot(2,1,2)
plot(fopt1)
xlabel('iterations')
ylabel('function minimum')

tic
% study 2- stalagmite behaviour-with upper and lower bounds
for i=1:num_cases
    [inputs,fopt2(i)]=ga(@stalagmite_function,2,[],[],[],[],[0,0],[0.6,0.6]);
    xopt2(i)=inputs(1)
    yopt2(j)=inputs(2)  
end
time2_study=toc
figure(2)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt2,yopt2,fopt2,'Marker','o','markersize',5,'markerfacecolor','r');
title('bounded inputs')
subplot(2,1,2)
plot(fopt2)
xlabel('iterations')
ylabel('function minimum')

tic
% study3-increaasing GA iterations with specified boundaries
for i=1:num_cases
    [inputs,fopt3(i)]=ga(@stalagmite_function,2,[],[],[],[],[0,0],[0.6,0.6],[],[],option);
    xopt3(i)=inputs(1)
    yopt3(j)=inputs(2)
end

time3_study=toc
figure(3)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt3,yopt3,fopt3,'Marker','o','markersize',5,'markerfacecolor','r');
title('increasing GA iterations')
subplot(2,1,2)
plot(fopt3)
xlabel('iterations')
ylabel('function minimum')



% Declaration of stalagmite function

function [stalagmite]=stalagmite_function(input_vector)
x=input_vector (1)
y=input_vector (2)
f1_x=(sin(5.1*pi*x+0.5)).^6;
f2_y=(sin(5.1*pi*y+0.5)).^6;
f3_x=exp((-4*0.6931).*(x-0.0667)^2/(0.64));
f4_y=exp((-4*0.6931).*(x-0.0667)^2/(0.64));
stalagmite=(f1_x.*f2_y.*f3_x.*f4_y)
end