Week 4.1 - Genetic Algorithm

Objective

• Write a code in MATLAB to optimise the stalagmite function and find the global maxima of the function.
• Clearly expalin the concept of genetic algorithm in your own words and also explain the syntax for ga in MATLAB.

Stalagmite function:

Genatic Algorithom:

The Genatic algorithm was created based on the concept of natural selection theory. Natural selection is a biological theory was first proposed by Charles Darwin. In natural selection theory the organisms with high ability is qualified to survive in the current environment and generates new organisms into the new generation.

GA Syntax:

set 1

x = ga(fun,nvars)

x = ga(fun,nvars,A,b)

x = ga(fun,nvars,A,b,Aeq,beq)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options)

set 2

[x, fval] = ga(any one of the set 1)

[x, fval,exitflag,output] = ga(any one of the set 1)

[x, fval,exitflag,output,population,score] = ga(any one of the set 1)

x - local minimum

fun - given function

nvars - number of variables

A,b,Aeq,beq - linear equalities and inequalities

lb,ub - lower bounds and upper bounds

nonlcon,intcon - nonlinear equalities and inequalities

options - it used to change the fixed function tolerance

fval, the value of the fitness function at x

Code

close
clear
clc


% All inputs

x = linspace(0,1,150);
y = linspace(0,1,150);
[xx, yy] = meshgrid(x, y);
number_of_cases = 150;

% function inetializing

f=ones(length(x), length(y));
fopt = ones(number_of_cases,number_of_cases);
xopt = ones(number_of_cases,number_of_cases);
yopt = ones(number_of_cases,number_of_cases);
xoptim = ones(number_of_cases,number_of_cases);
yoptim = ones(number_of_cases,number_of_cases);
foptim = ones(number_of_cases,number_of_cases);
fop = ones(number_of_cases,number_of_cases);
xop = ones(number_of_cases,number_of_cases);
yop = ones(number_of_cases,number_of_cases);





for i = 1 : length(xx)
    for j = 1 : length(yy)
        input(1) = xx(i,j);
        input(2) = yy(i,j);        
        f(i, j) = fun(input);
    end
end

figure(1)
surface(xx, yy, -f)
shading interp
xlabel('x value')
ylabel('y value')
zlabel('valures of staligmate function')
[output, fval] = ga(@fun, 2);



% case 1


for i = 1 : number_of_cases
     [inputs , fopt(i)] = ga(@fun, 2);
     xopt(i) = inputs(1);
     yopt(i) = inputs(2);     
end


% plotting

figure(2)
subplot(2,1,1)
hold on 
surfc(x,y,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','*')
xlabel('x value')
ylabel('y value')
zlabel('valures of staligmate function')
title('Case 1')
subplot(2,1,2)
plot(-fopt)
xlabel('cases(iterations)')
ylabel('Maximum values')


% case 2

for i = 1 : number_of_cases
    [in, foptim(i)] = ga(@fun,2,[],[],[],[],[0;0],[1;1]);
    xoptim(i) = in(1);
    yoptim(i) = in(2);
end

% plotting

figure(3)
subplot(2,1,1)
hold on 
surfc(x,y,-f)
shading interp
xlabel('x value')
ylabel('y value')
zlabel('valures of staligmate function')
plot3(xoptim,yoptim,-foptim,'marker','*')
title('Case 2')
subplot(2,1,2)
plot(-foptim)
xlabel('cases(iterations)')
ylabel('Maximum values')


% case 3

options = optimoptions('ga');
options = optimoptions(options,'populationsize',300);

for i = 1 : number_of_cases
    [inp, fop(i)] = ga(@fun,2,[],[],[],[],[0;0],[1;1],[],[], options);
    xop(i) = inp(1);
    yop(i) = inp(2);
end

% plotting

figure(4)
subplot(2,1,1)
hold on 
surfc(x,y,-f)
shading interp
plot3(xop,yop,-fop,'marker','*')
xlabel('x value')
ylabel('y value')
zlabel('valures of staligmate function')
title('Case 2')
subplot(2,1,2)
plot(-fop)
xlabel('cases(iterations)')
ylabel('Maximum values')



% function

function [F] = fun(input)


F1x = (sin(5.1*pi*input(1)+.5)).^6;

F2x = exp(-4*log(2)*((input(1)-.0667).^2./.64));

F1y = (sin(5.1*pi*input(2)+.5)).^6;

F2y = exp(-4*log(2)*((input(2)-.0667).^2./.64));

F = -(F1x.*F2x.*F1y.*F2y);
end

Code explanation:

• First clear all previous things
• giving input variables
• then initialises function with ones matrix
• do some required function call for create graph
• then firstly create a basic ga without any iterations and with 2 variables
• then create cases and create plots attaced to it
• after all these required function is defined at the last

Results:

• 
• 
• In this first case giving 150 itrations to "ga" and not giving any initial and final points. so we get a large number of values
• 
• At second case we provide limiting values then we get more acurate values we need
• 
• In the third case we change options and increase the population size into 300 and so that we get more acurate results.

Reference:

1. An Article by Musatafa Abbas Albadr, Sabrina Tiun, Masri Ayob and Fahad AL-Dhief. "Genetic Algorithm Based on Natural Selection Theory for Optimization Problems".
2. ga