Week 4.1 - Genetic Algorithm

AIM:

To write a code in MATLAB to optimise the stalagmite function and find the global maxima of the function and to explain the syntax for generic algorithm in MATLAB.

OBJECTIVE:

• To write a code in MATLAB to optimise the stalagmite function and find the global maxima of the function.
• To expalin the concept of genetic algorithm and also explain the syntax for ga in MATLAB in your report.
• To plot graphs for all 3 studies and for F maximum vs no. of iterations. Title and axes labels are a must, legends could be shown if necessary.

WHAT IS GENETIC ALGORITHM ?

A genetic algorithm is a search heuristic that is inspired by Charles Darwin’s theory of natural evolution. This algorithm reflects the process of natural selection where the fittest individuals are selected for reproduction in order to produce offspring of the next generation.

NOTION OF NATURAL SELECTION

The process of natural selection starts with the selection of fittest individuals from a population. They produce offspring which inherit the characteristics of the parents and will be added to the next generation. If parents have better fitness, their offspring will be better than parents and have a better chance at surviving. This process keeps on iterating and at the end, a generation with the fittest individuals will be found.

This notion can be applied for a search problem. We consider a set of solutions for a problem and select the set of best ones out of them.

Five phases are considered in a genetic algorithm.

1. Initial population
2. Fitness function
3. Selection
4. Crossover
5. Mutation

1. Initial Population

The process begins with a set of individuals which is called a Population. Each individual is a solution to the problem you want to solve.

An individual is characterized by a set of parameters (variables) known as Genes. Genes are joined into a string to form a Chromosome (solution).

In a genetic algorithm, the set of genes of an individual is represented using a string, in terms of an alphabet. Usually, binary values are used (string of 1s and 0s). We say that we encode the genes in a chromosome.

2. Fitness Function

The fitness function determines how fit an individual is (the ability of an individual to compete with other individuals). It gives a fitness score to each individual. The probability that an individual will be selected for reproduction is based on its fitness score.

3. Selection

The idea of selection phase is to select the fittest individuals and let them pass their genes to the next generation.

Two pairs of individuals (parents) are selected based on their fitness scores. Individuals with high fitness have more chance to be selected for reproduction.

4. Crossover

Crossover is the most significant phase in a genetic algorithm. For each pair of parents to be mated, a crossover point is chosen at random from within the genes.

5. Mutation

In certain new offspring formed, some of their genes can be subjected to a mutation with a low random probability. This implies that some of the bits in the bit string can be flipped.

MATHEMATICAL EQUATION

PROGRAM

Plotting global maxima

clear all 
close all
clc

% Defining our search space
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);

% Creating a 2Dimensional mesh
[xx yy] = meshgrid(x,y);

% Evaluating 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

surfc(xx,yy,f)
xlabel('x-value')
ylabel('y-value')

shading interp

[inputs,fval] = ga(@stalagmite,2)

Function

function[f] = stalagmite(input_vector)

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

f1 = (sin(5.1*pi*x+0.5))^6;
f2 = (sin(5.1*pi*y+0.5))^6;
f3 = exp(-4*log(2)*(x-0.667)^2/0.64);
f4 = exp(-4*log(2)*(y-0.667)^2/0.64);

f = (f1*f2*f3*f4);
end

Result:

FOR PLOTTING GLOBAL MINIMA AND FINDING THE MINIMUM OF THE FUNCTION GA IS USED

clear all
close all
clc

x=linspace(0,0.6,150)
y=linspace(0,0.6,150)

[x,y]=meshgrid(x,y)

num_cases=50
f=get_stalagmite(x,y)


tic

%study 1 - statistical behavior
for i=1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2)
    xopt(i)=inputs(1)
    yopt(i)=inputs(2)
end

study1_time=toc

%plotting1
figure(1)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfacecolor','g')
title('Unbounded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('function minimum')

tic

%study 2 -statistical behaviour

for i=1:num_cases
 [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[0.4;0.4])
    xopt(i)=inputs(1)
    yopt(i)=inputs(2)
end

study2_time=toc

%plotting2

figure(2)

subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','o','markersize',3,'markerfacecolor','g')
title('Bounded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('function minimum')

%increrasing ga iteration
options=optimoptions('ga')
options=optimoptions(options,'populationsize',170)

tic

for i=1:num_cases
 [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[0.4;0.4],[],[],options)
    xopt(i)=inputs(1)
    yopt(i)=inputs(2)
end

study3_time=toc

%plotting3

figure(3)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfacecolor','g')
title('Bounded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('function minimum')

 % Function

function [f]=get_stalagmite(x,y)

f1=(sin(5.1*pi.*x+0.5)).^6;
f2=(sin(5.1*pi.*y+0.5)).^6;
f3=exp(-4*log(2).*(x-0.0667).^2/0.64);
f4=exp(-4*log(2).*(y-0.0667).^2/0.64);

f=-(f1.*f2.*f3.*f4);
end

RESULTS:

REFERENCES:

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

https://www.hindawi.com/journals/mpe/2015/212794/