Week 4.1 - Genetic Algorithm

Genetic Algorithm

Aim:

• To optimize the stalagmite function using genetic algorithm techniques using Matlab.

• Fitness function: It is the function that GA is trying to optimize.
• A population of chromosomes: It is the numerical values that represent all the solutions to the function. The selection operator chooses some of the chromosomes (with replacement) for reproduction based on probability distribution. the fitter the chromosome more likely it is to be selected.
• Cross over: This operator swaps in random a subsequence of two of the chosen chromosomes to create two offspring. Usually, each chromosome is converted into binary code and then cross-over is done from one parent chromosome into another.
• Mutation: The mutation operator randomly flips individual bits in the new chromosomes.

• They usually perform better than traditional feature selection techniques.
• Genetic algorithms can manage data sets with many features.
• They don't need specific knowledge about the problem under study.
• These algorithms can be easily parallelized in computer clusters.

1. The algorithm begins by creating a random initial population. The algorithm then creates a sequence of new populations.
2. At each step, the algorithm uses the individuals in the current generation to create the next population.

• Scores each member of the current population by computing its fitness value. These values are called the raw fitness scores.
• Scales the raw fitness scores to convert them into a more usable range of values. These scaled values are called expectation values.
• Selects members, called parents, based on their expectations.
• Some of the individuals in the current population that have lower fitness are chosen as elite. These elite individuals are passed on to the next population.
• Produces children from the parents. Children are produced either by making random changes to a single parent mutation or by combining the vector entries of a pair of parents crossover.
• Replaces the current population with the children to form the next generation. The algorithm stops when one of the stopping criteria is met.

CODE:

Function code:

function [f] = stalagmite(input_vector)
x = input_vector(1);
y = input_vector(2);
f1_x = (sin((5.1*pi*x)+0.5)).^6;
f1_y = (sin((5.1*pi*y)+0.5)).^6;
f2_x = exp(((-4*log(2))*(x-0.0667).^2)/0.64);
f2_y = exp(((-4*log(2))*(y-0.0667).^2)/0.64);
[f] = -(f1_x.*f1_y.*f2_x.*f2_y);
end

Main code:

% A program to genetic Genetic algorithm
clear all
close all
clc


% Defining our search space
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);
num_cases = 80;
% Creating a 2D 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_cases;
    x_1(k)=input_vector(1);
    y_1(k)=input_vector(2);
  
end
study1_time = toc

tic
for s = 1:num_cases;
    [input,fval(s)] = ga(@stalagmite,2,[],[],[],[],[0,0],[.6,.6]);
    x_2(s) = input(1);
    y_2(s) = input(2);
end
Study2_time = toc

% 3rd case study increasing ga iterations
options = optimoptions('ga');
options = optimoptions(options,'populationsize',150);
tic
for m = 1:num_cases;
    [input,fval(m)] = ga(@stalagmite,2,[],[],[],[],[0,0],[.6,.6],[],[],options);
    x_3(m) = input(1);
    y_3(m) = input(2);
end
study3_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','g')
title('unbounded inputs')
subplot(2,1,2)
plot(-fval)
xlabel('iterations')
ylabel('function minimum')

% 2nd case study statistical behavior
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','g')
title('unbounded inputs')
subplot(2,1,2)
plot(-fval) 
xlabel('iterations')
ylabel('function minimum')

% 3rd case study increasing ga iterations
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','g') 
title('unbounded inputs')
subplot(2,1,2)
plot(-fval)
xlabel('iterations')
ylabel('function minimum')

Explanation of the code:

• The code starts with defining x and y as inputs using linspace.
• A function known as Stalagmite_func is defined as shown in the 2nd stage of codes.
• 2 for loops are incorporated in the main code so that all the combinations of x and y values are taken in the stalagmite_func function
• In the 1st study, the GA function runs for 80 iterations. Plots of the stalagmites and maximum values generated in each iteration are shown.
• In the 2nd study, the GA function is incorporated with lower and upper bounds and runs for 80 iterations. Plots of stalagmites and maximum values generated in each iteration are shown.
• In the 3rd study, the GA function is incorporated with lower and upper bounds along with the population size of 200. Plots of the stalagmites and maximum values generated in each iteration are shown.

RESULT: