Challenge

Week 5 - Genetic Algorithm

SShailendra Shankar18 Apr 2023

AIM: To write a code in MATLAB to optimize the stalagmite function and find the global maxima of the function. To explain the concept of genetic algorithm and also the syntax for ga in MATLAB. To plot the graphs for all 3 studies and for F maximum vs no. of iterations.

Theory: Stalagmites are rocks that rise from the floor of a cave due to the accumulation of materials from drops from the ceiling. The shape of these stalagmites are the basis for calling these programs as "STALAGMITES".

Genetic Algorithm is a method based on natural selection, the basis of biological evolution, for addressing both limited and unconstained optimization problems. The population size of distinct solution is always changing thanks to evolutionary algorithm.

Phases of Genetic Algorithm:

Initialization of Population: Each gene stands for a parameter in the response. The chromosome is this set of variables that makes up the solution.
Fitness Function: Out of the available chromosomes, we must choose the finest ones to create offspring, hence each chromosome is assigned a fitness value.
Selection: Finding the area where getting the greatest answer is more likely the major objective of theis phase.
Reproduction: Generation of offsprings happen by two ways:
- Crossover: Crossover is the most vital stage in the Genetic Algorithm. During a crossover a random point is selected while mating a pairof parents, to generate offspring.
- Mutation: Some of the newly produced offsprings may be subjected to low probability mutation. This suggests that some bits of the chromosome can be switched around. Mutation occurs to maintain population variety and to prevent premature convergence.
Convergence (When to stop): Following are the conditions which determine when to stop:
- when a predetermined number of generations have passed and there is no improvement in the quality of the answer.
- when a fixed number of generations or years have passed.
- until a workable resolution is found.

Syntax for Genetic Algorithm:

[x,fval,exitflag,output,population,scores]=ga(func,nvars,A,b,Aeq,beq,lb,ub,nonIcon,IntCon,options)

where,

x - optimum solution
exitflag - reason Genetic Algorithm stopped
output - information about optimization process
population - final population
scores - final scores
func - fitness function
nvars - number of variables
A,b - linear contraints
Aeq,beq - linear equality contraints
lb,ub - lower and upper bounds (limits)
nonIcon - use of non-linear constraints
IntCon - to get solution in integer
options - to set the population size with non-default options

Function Code:

function [f] = stalagmite(input_vector)
x = input_vector(1);
y = input_vector(2);
f1x = (sin(5.1*pi*x + 0.5))^6;
f1y = (sin(5.1*pi*y + 0.5))^6;
f2x = exp((-2.7726)*(x-(0.0667))^2/0.46);
f2y = exp((-2.7726)*(y-(0.0667))^2/0.46);

f = -(f1x*f1y*f2x*f2y);
end

Explanation:

function to perform the task of stalagmite function
initialize first column
initialize second column
initialize terms according to equation
calculation the value according to the stalagmite function

Main Code:

clear all
close all
clc

%Defining our search space
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);
num_case = 60;

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

%Evaluating the 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

tic
for k = 1:num_case;
    [inp, fval(k)] = ga(@stalagmite,2);
    x_1(k) = inp(1);
    y_1(k) = inp(2);
end
study1_time = toc

figure (1)
hold on
grid on
surfc(xx,yy,-f)
shading interp
title('stalagmite plot')
xlabel('X')
ylabel('Y')

figure(2)
subplot(2,1,1)
surfc(xx,yy,-f)
shading interp
hold on
plot3(x_1,y_1,-fval,'marker','o','MarkerSize',5,'MarkerFaceColor','m')
title('unbounded inputs')
subplot(2,1,2)
plot(-fval)
xlabel('iterations')
ylabel('function minimum')


% 2nd case study statistical behaviour
tic
for s = 1:num_case;
    [input,fval(s)]= ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6]);
    x_2(s)= input(1);
    y_2(s)= input(2);

end
study2_time = toc;
figure(3)
subplot(2,1,1)
surfc(xx,yy,-f)
shading interp
hold on
plot3(x_2,y_2,-fval,'marker','o','MarkerSize',5,'MarkerFaceColor','m')
title('unbounded inputs')
subplot(2,1,2)
plot(-fval)
xlabel('iterations')
ylabel('function minimum')

% 3rd case study increasing ga iterations
options = optimoptions('ga');
options = optimoptions(options,'PopulationSize',500);

tic
for m = 1:num_case;
    [input,fval(m)]= ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6],[],[],options);
    x_3(m)= input(1);
    y_3(m)= input(2);

end
study3_time = toc
figure(4)
subplot(2,1,1)
surfc(xx,yy,-f)
shading interp
hold on
plot3(x_3,y_3,-fval,'marker','o','MarkerSize',5,'MarkerFaceColor','m')
title('unbounded inputs')
subplot(2,1,2)
plot(-fval)
xlabel('iterations')
ylabel('function minimum')

Explanation: