Week 4.1 - Genetic Algorithm

AIM:

To write a programme optimise the stalagmite function and find the the global maxima of the function by using genetic algorithm.

THEORY:

GENETIC ALGORITHM:

• Genenatic algorithm was based on the principle of darwin's theory of evolution which states that survival of the fittest.
• Here,this algorithm is mainly used to evaluate the solutions of an objective function by considering the random population.Here a fitness function is usually introduction to find the fittest of the solution.This method initially find the parent solution which is termed as the chromosomes,which inturn will undergo a fitness function to find the suitable of solution.Now,set of chromosomesgets mutated by the process of crossover and produce new chromosomes which will be combinations of parent genes.
• Mutation is a process of replacing random genes with another set of random genes.The rate of both crossover and mutation rate value.After several generation the solution value converges to a value which is regarded as the best solution
• Genetic algorithm generally find the local manima which is found to vary each time we run the programme.The consistancy of this can be improved by,Increasing the population size,Assigning the boundedness,Changing the mutation and Crossover rate etc..,

GENERAL SYNTAX OF GENETIC ALGORITHM:

Var_x=ga(@function,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options).

• @function:The function to be analysed to find the minima
• nvars:This represents the number of the variables to be evaluated
• A,b:This represents the linear inequalities contraints
• Aeq,beq:This represents the linear equalities
• lb,ub:Represents the lower and upper bound values in which the function is to evaluted
• nonlcon:This shows the non linear constraints
• intcon:This represents the integer constraints
• options:These are the option necessary for the algorithm such as population size,crossover function etc to get consistent values.

STOPPING CRITERIA:

In an effort to reduce the lengthy computational time of algorithms, the number of the unique determined schedule is selected as an alternated algorithm stopping criteria condition to identify the trade-off between algorithm stopping criteria and algorithm performance measures.

STALAGMITE FUNCTION (GOVERNING EQUATION):

f(x,y)=f1,xf2,xf1,yf2,y

`f1,x=[sin(5.1pix+0.5)]^6`

`f1,y=[sin(5.1piy+0.5)]^6`

`f2,x=exp[-4ln(2).((x-0.0667)^2/0.64)]`

`f2,y=exp[-4ln(2).((y-0.0667)^2/0.64)]`

STALAGMITE FUNCTION:

function[f]=stalagmite(input_vector);
x=input_vector(1);
y=input_vector(2);

f1=(sin(5.1.*pi.*x+0.5)).^6;
f2=(sin(5.1.*pi.*y+0.5)).^6;
f3=exp(-4*log(2).*(x-0.667).^2/0.64);
f4=exp(-4*log(2).*(y-0.667).^2/0.64);
[f]=-(f1.*f2.*f3.*f4)
end

MAIN CODE:

PROGRAM-1

%MATLAB progrma to find maxima value for stalagmite function GA:
clear all
close all
clc

%defining our search space:
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
num_cases = 50
%creating 2-dimensional 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
surf(x,y,-f)
xlabel('x value stalagmite')
ylabel('y value stalagmite')
shading interp


[x,fval] = ga(@stalagmite,2);

EXPLANATION:

1.The code is started with the three comands.

clear all-clear the workspace

close all-clear the figure window

clc-clear the command window

2.We define the value of x and y using the linspace().

3.A meshgrid [xx yy] is created using x and y matrix.

4.Then we evaluate the stalagmite function running a for-loop and here we call the stalagmite function.

5.plot the stalagmite function in as a 3D surface.

study of genetic algorithm for the random sampling:

• we run to find the maxima value by running the ga() for 50 times.
• Also we calculate the running time.
• Then we plot the stalagmite function along the maximum values which is we have got from multiple iterations,also we plot function maxima vs iteraction on the same figure.

6.Tic and Toc :Tic works with the toc function to measure the elapsed time.The 'Tic' function records the current function and the 'Toc' function uses to record value to calculate the elapsed.

PROGRAM-2

%case 1:finding maxima without giving any constraints or unbounded value
clear all
close all
clc

%defining our search space:
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);

%creating 2-dimensional 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
surf(x,y,-f)
xlabel('x value stalagmite')
ylabel('y value stalagmite')
shading interp

tic
%statical behaviour without bounds:
num_cases = 50
for k=1:num_cases
    [inputs,fopt(k)]= ga(@stalagmite,2);
    xopt(k)=inputs(1);
    yopt(k)=inputs(2);
end
study_time1=toc
figure(1)
subplot(2,1,1)
hold on 
surfc(xx,yy,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','MarkerSize',5,'MarkerFaceColor','r')
title('unbounded inputs')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('function minimize')

EXPLANATION:

Study of genetic algorithm without any constraints or boundedness

• Now here we compute the maxima by using genetic algorithm.
• The syntax of GA include calling the function whose maxima is to be found and the no.of.variables,here begin the case one we do not define the popoulation nor the bounds.
• Since,GA return to maxima value we multiple the results of stalagmite function with a negative sign.
• So the GA return to the maximum value with the negative sign,thus the point of incident of maximum value and the value itself is also stored on to an array.
• The results of stalagmite function is popltted as a 3-Dimensional array and the maxima along with their point of incident is plotted.

PROGRAM-3

%case 2:finding maxima with giving constraints and bounded values
clear all
close all
clc

%defining our search space:
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);

%creating 2-dimensional 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
surf(x,y,-f)
xlabel('x value stalagmite')
ylabel('y value stalagmite')
shading interp

tic
%statical behaviour with upper and lower bounds:
num_cases = 50
for k=1:num_cases
    [inputs,fopt(k)]= ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6]);
    xopt(k)=inputs(1);
    yopt(k)=inputs(2);
end
study_time1=toc
figure(2)
subplot(2,1,1)
hold on 
surfc(xx,yy,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','MarkerSize',5,'MarkerFaceColor','r')
title('bounded inputs')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('function minimize')

EXPLANATION:

• Here in the case two we include the bounds which is describes the domain in which the GA has to compuate the maxima.
• The bounds were defined as 0-0.6 for both x and y.The syntax for which is shown below

%case 3: Increasing the GA iterations
clear all
close all
clc

%defining our search space:
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);

%creating 2-dimensional 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
surf(x,y,-f)
xlabel('x value stalagmite')
ylabel('y value stalagmite')
shading interp

%statical increasing number of iterations
options = optimoptions('ga')
options = optimoptions(options,'populationsize',170)

tic
num_cases = 50
for k=1:num_cases
    [inputs,fopt(k)]= ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6],[],[],options);
    xopt(k)=inputs(1);
    yopt(k)=inputs(2);
end
study_time3=toc
figure(3)
subplot(2,1,1)
hold on 
surfc(xx,yy,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','MarkerSize',5,'MarkerFaceColor','r')
title('optimization')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('function minimize')