Week 4.1 - Genetic Algorithm

Aim: 

        To write a progarm for Stalagmite Mathemitical Model &  Genetic Algorithm. 

%Stalagmite Mathematical model 

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 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)=functionofstalagmite(input_vector); 

end 

end 

  

surfc(xx,yy,-f); 

shading interp 

xlabel('xvalue'); 

ylabel('yvalue'); 

  

numspaces=50; 

  

%Study1-statistical behavior 

for i=1:numspaces 

    [inputs,fopt(i)]=ga(@functionofstalagmite,2); 

    xopt(i)=inputs(1); 

    yopt(i)=inputs(2); 

end 

  
studytime=tic; 


figure(1) 

subplot(2,1,1) 

hold on 

surfc(x,y,-f) 

  
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r') 

title('Unbound inputs') 

subplot(2,1,2) 

plot(-fopt) 

shading interp 

xlabel('Iteration') 

ylabel('function minimum') 

tic 

  

%Study2-statistical behavior with upper & lower limits 

for i=1:numspaces 

    [inputs,fopt(i)]=ga(@functionofstalagmite,2,[],[],[],[],[0;0],[1:1]); 

    xopt(i)=inputs(1); 

    yopt(i)=inputs(2); 

end 

  
study2_time=toc;
 

figure(2) 

subplot(2,1,1) 

hold on 

surf(x,y,-f) 

shading interp 

plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r') 

title('bounded inputs') 

subplot(2,1,2) 

plot(-fopt) 

  

%Study3-increasing GA iterations 

options=optimoptions('ga'); 

options=optimoptions(options,'populationsize',500); 

tic 

  
for i=1:numspaces 

    [inputs,fopt(i)]=ga(@functionofstalagmite,2,[],[],[],[],[0;0],[1:1],[],[],options); 

     

    xopt(i)=inputs(1); 

    yopt(i)=inputs(2); 

end 

study3_time=toc 

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) 

  

  
%function of stalagmite 

function f=functionofstalagmite(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(-4*log(2)* (x-0.0667)^2/0.64); 

f2y=exp(-4*log(2) *(y-0.667)^2/0.64); 

f=-f1x*f1y*f2x*f2y; 

 end 

Objective: 

                To plot graphs for  Stalagmite Mathemitical Model &  Genetic Algorithm. 

Theory of GA: 

A genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms . Genetic algorithms are commonly used to generate high-quality solutions to optimization and search problems. In a genetic algorithm, a population of candidate solutions to an optimization problem is evolved toward better solutions The evolution usually starts from a population of randomly generated individuals, and is an iterative process, with the population in each iteration called a generation. The algorithm terminates when either a maximum number of generations. A GA proceeds to initialize a population of solutions and then to improve it through repetitive application of the mutation, crossover, inversion and selection operators. 

Syntax:

1.x = ga(fun,nvars)

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

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

Description of Syntax:

1.x = ga(fun,nvars):

 It finds a local unconstrained minimum, x, to the objective function, fun. nvars is the dimension (number of design variables) of fun. In this x is the input,ga is genetic algorithm , fun is the function & nvars is the dimension.

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

This Syntax defines that a set of lower and upper bounds on the design variables, x, so that a solution is found in the range lb ≤ x ≤ ub. (Set Aeq=[] and beq=[] if no linear equalities exist.) . In this syntax x is the input,ga is genetic algorithm, A & B are the Linear Inequality constraints, Aeq & beq are Linear equality constraints, lb & ub are the lower bounds & upper bounds respectively.

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

   This requires that the variables listed in intcon take integer values. (Set nonlcon=[] if no nonlinear constraints exist.) .In this syntax x is the input,ga is genetic algorithm, nvars is the dimension, A & B are the Linear Inequality constraints, Aeq & beq are Linear equality constraints, lb & ub are the lower bounds & upper bounds respectively.nonlcon is Non linear constraints, intcon is integer constraints, & options is optimization

Explanation of Output:

1.The first output or graph,plots the x & y values of Stalagmite Mathematical Model.

2.The second output or graph is the Study 1 of GA. In this, the plot mentions the unbounded outputs  X, Y and  Z values are plotted in the output

3.Whereas in the third output is th study 2 of GA.  In this, the plot shows us the bounded outputs   X, Y and  Z values are plotted in the output

4.Both the 2nd and 3rd plots are ununiform, its not in a uniform way.

5.The fourth output or plot depends on the No. of population, as per population size the plotting has been done & this is also a bounded plot. In this  also X,Y & Z values are plotted, comparing to 2nd and 3rd plot is those both plots ununiform , whereas  the 4 plot is in a uniform way.