Week 4.1 - Genetic Algorithm

AIM:

To optimise the stalagmite function and find the global maxima of the function using MATLAB

Objectives:

To explain the concept of genetic algorithm and its syntax and also find the global maxima of the function 
Genetic Algorithm
A genetic algorithm (GM) is a method for solving both constrained and unconstrained optimization problem based on a natural selection process that mimics biological  evolution the algorithm repeatedly modifies a population of individual solution at each step

GA SYNTAX:

The genetic algorithm use three main type of rules at each step to create the next generation from the current population
1]  Selection rule select the individuals called parents that contribute to the population at the next generation 
2]  Crossover rule combine two parents two from children for the next generation 
3]  Mutation rules apply random changes to individual parents to from children
•	X = ga(fun,nvars)
•	X = ga (fun,nvars ,A,b)
•	X = ga(fun,nvars,A,b,Aeq,beq)
•	X = ga(fun,nvars,A,b,beq,lb,ub)
•	X = ga(fun,nvars,A,b,beq,lb,ub,nonlcon)
•	X = ga((fun,nvars,A,b,beq,lb,ub,nonlcon,option)
•	X = ga(fun,nvars,A,b,beq,lb,ub,nonlcon,intcon)
•	X = ga(problem)
•	[x,fval] = ga(_)
•	X,fval,exiting,output] = ga(_)

Working of GA:

1]  the algorithm being by creating a random initial population 
2]  scales the raw fitness scores to convent them into a more usable range of values 
3]  some of the individuals in the current population that have fitness are chosen at elite
4]  children are produced either by marking random changes
5]  the algorithm stops when one of the stopping criteria is met

Stopping Criteria:

1] Stopping criteria determines what cause the algorithm to terminates  
2] Generation specifies the maximum number of iteration genetic algorithm performs
3] time limit specifies the maximum time in second the genetic algorithm runs before stopping
•Fitness limit 
•Stall generation 
•Stall time limit
•Stall test 
•Function tolerance
•Constraint tolerance


% stalagmite function programing

function [f]=func(input_vector)
a1=(sin(5.1*pi*input_vector(1)+0.5))^6;
a2=(sin(5.1*pi*input_vector(2)+0.5))^6;
a3=exp(-4*log(2)*((input_vector(1)-0.0667)^2)/(0.64));
a4=exp(-4*log(2)*((input_vector(2)-0.0667)^2)/(0.64));
f=-(a1*a2*a3*a4);
end

% code for all three studies 
close all
clear all 
clc
% input
x = linspace(0,0.6,150)
y = linspace(0,0.6,150)
num_cases = 60
%storing in 2d arry
[xx,yy] = meshgrid(x,y);
% calling the 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) = func(input_vector);
    end
end
% study 1 behaviour
tic
for i = 1:num_cases
    [inputs,fopt(i)] = ga(@func,2);
    xopt(i) = inputs(1);
    yopt(i) = inputs(2);
end
studytime1 = toc 
% unbounded inputs
 figure (1)
 subplot(2,1,1)
 hold on
 surfc(x,y,-f)
 shading interp
 plot3(xopt,yopt,-fopt,'Marker','o','MarkerSize',5,'MarkerFaceColor','r')
 title('unbounded inputs')
 subplot(2,1,2)
 plot(-fopt)
 xlabel('interations')
 ylabel('function maximum')

 % study 2 behaviour
 tic
 for i=1:num_cases
     [inputs,fopt(i)] = ga(@func,2,[],[],[],[],[0,0],[0.5,0.5]);
     xopt(i) = inputs(1);
     yopt(i) = inputs(2);
 end
 studytime2 = toc
 %bounded inputs
 figure(2)
 subplot(2,1,1)
 hold on
 surfc(x,y,-f)
 shading interp
 plot3(xopt,yopt,-fopt,'Marker','o','Markersize',5,'MarkerFaceColor','r')
 title('bounded inputs')
 subplot(2,1,2)
 plot(-fopt)
 xlabel('iteration')
 ylabel('function maximu


  % study 3 behaviour
 tic
 options = optimoptions('ga')
 options = optimoptions(options,'PopulationSize',200);
 for i=1:num_cases
     [inputs,fopt(i)] = ga(@func,2,[],[],[],[],[0;0],[0.5;0.5],[],[],options);
     xopt(i) = inputs(1);
     yopt(i) = inputs(2);
 end
 studytime3 = toc
 % optimisation
 figure(3)
 subplot(2,1,1)
 hold on
 surfc(x,y,-f)
 shading interp
 plot3(xopt,yopt,-fopt,'Marker','o','Markersize',5,'MarkerFaceColor','r')
 title('bounded inputs')
 subplot(2,1,2)
 plot(-fopt)
 xlabel('iteration')
 ylabel('function maximum')