Challenge

Week 4.1 - Genetic Algorithm

AArpit Hardiya14 Aug 2022

Aim:-

To explain the Concept of Ga and syntax for Ga in MATLAB.
To write a code in MATLAB to optimize the stalagmite function and find the global maxima of the function.
On several execution's code should provide the same output each time.
To plot a graph for all 3 studies and for F maximum vs no. of iterations. Title and axes labels are a must, legends could be shown if necessary.

Concept of Genetic Algorithm:-

Genetic algorithm is a strategies that guide the search process inspired by the process of natural selection. Genetic algorithm are commonly used for generating high quality solutions to optimization and search problems by relaying on biologically inspired operators such as mutation, crossover and selection.

Initial population

The initial population is generated randomly which later evolved to the optimum solution.

Calculate the fitness function

The best solution is selected for the next generation based on the fitness function, in our case our fitness function is given as

We simplified the above code below to use it in matlab as a function.

function [stalagmite]=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*0.6931).*(x-0.00667)^2/(0.64));

f2_y=exp((-4*0.69310).*(y-0.00667)^2/(0.64));

stalagmite= (f1_x.*f2_x.*f1_y.*f2_y);

end

Selection

During each successive generation, a portion of the existing population is selected to breed a new generation. This selection of this best solution depends upon fitness function

Mutation

Mutation is used to maintain and introduce the diversity in the genetic population with small random tweaks in the previous solutions to create new better solutions.

Types of mutation operators.

Bit Flip Mutation
Random Resetting
Swap Mutation
Scramble Mutation
Inversion Mutation

Crossover

One point cross over
Tow point cross over
Uniform cross over

Termination criteria

Ga in the beginning is very fast creating better solution with each iteration but after some time he process become saturated in later stages where the improvement in solution is minimal and we need to stop the process at the point where the solution is very close to optimal.

Termination conditions are:-

When no improvement In solution is observed for some iterations (pre-defined )
When the absolute number of generations is reached
Algo. terminated when the counter reaches the predetermined value.

Syntax we used for Ga

X=ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonIcon,options)

X	best point ga located during iteration
Fun	fitness function
Nvars	number of variables
A	Liner inequality constraints specified as a real matrix.
b	Liner inequality constraints specified as real vector.
Aeq	Liner Equality constraints specified as real matrix.
beq	Liner Equality constraints specified as real vector.
lb	Lower bound
ub	Upper bound
nonIcon	nonlcon is a function that accepts a vector or array x and returns two arrays, c(x) and ceq(x).
Options	Options created using optimoptions or an options structure.

BODY OF CODE

Creating the fitness function to find the global maxima.

function code

negative sign used to find the maxima, because by default in matlab GA finds the minima only.

MAIN BODY OF THE CODE (CODE TO PLOT GA)

Creating the 3d space with use of meshgrid command to find the GA the maxima and minima in that range.

clear all 
close all 
clc % clear all close all and clc to clear older data

x=linspace(0,0.6,150); %creating the search space for the ga in between it plot the graph
y=linspace(0,0.6,150);
 
[xx,yy]=meshgrid(x,y);% creating meshgrid for 3 dimensionl plot 
for i=1:length(xx); % nested for loop for processing each valu of xx and yy at a time
    for j=1:length(yy);
        input_vector(1)=xx(i,j); % defining the cordinates for ploting the graph and this cordinates will be used in stalagmite as a input 
        input_vector(2)=yy(i,j);
        f(i,j) = stalagmite(input_vector);% calling the stalagmite function.
    end
end
surf(xx,yy,f); %creating the 3D graph 
shading interp % command to remove the mesh line from the graph 
xlabel('X axis')
ylabel('Y axis')
zlabel('Globel maxima/Minima ')

Out put of the above code

Genetic algoritm study 1

Defining the number of itiration for the genetic alogrithm

clear all 
close all 
clc % clear all close all and clc to clear older data

x=linspace(0,0.6,150); %creating the search space for the ga in between it plot the graph
y=linspace(0,0.6,150);
 
[xx,yy]=meshgrid(x,y);% creating meshgrid for 3 dimensionl plot 
for i=1:length(xx); % nested for loop for processing each valu of xx and yy at a time
    for j=1:length(yy);
        input_vector(1)=xx(i,j); % defining the cordinates for ploting the graph and this cordinates will be used in stalagmite as a input 
        input_vector(2)=yy(i,j);
        f(i,j) = stalagmite(input_vector);% calling the stalagmite function.
    end
end
num_cases=50% defining the number of itiration 
tic  %tic and toc command to find the study solving time 
for k=1:num_cases;  %for loop for solving the each value for 50 itiration 
    [input,fopt(k)]=ga(@stalagmite,2);
    xopt(k)=input(1)
    yopt(k)=input(2)
end
surf(xx,yy,f);% ploting the 3D graph 
shading interp %remove mesh line from the graph
study_time=toc %record the study time 
figure(1) %figure 1
subplot(2,1,1)%creating subplot on figure  
hold on% hold the previous graph 
surfc(x,y,f)%creating the graph
shading interp 
plot3(xopt,yopt,fopt,'marker','o','markersize',5) %ploting the 3D graph
title('unbounded value')% title 
xlabel('X axis')% defining axis 
ylabel('Y axis')
zlabel('Globel maxima/Minima ')
subplot(2,1,2)
plot(fopt)% ploting 2d plot for number of itirations and findings of maxima 
xlabel('Iteration')
ylabel('Function Maximum')

Graphical results of the above study.

time taken for the study is 34 sec

Genetic algoritm study 2

Reason for this Study 2

Despite of defining the search the values of maxima and minima is out of the search space so we need to limit the area by applying the limits to the search areas .

Bounding the area of search for GA by the use of uppr bound and lower bound

Code for the bounded GA

clear all 
close all 
clc % clear all close all and clc to clear older data

x=linspace(0,0.6,150); %creating the search space for the ga in between it plot the graph
y=linspace(0,0.6,150);
 
[xx,yy]=meshgrid(x,y);% creating meshgrid for 3 dimensionl plot 
for i=1:length(xx); % nested for loop for processing each valu of xx and yy at a time
    for j=1:length(yy);
        input_vector(1)=xx(i,j); % defining the cordinates for ploting the graph and this cordinates will be used in stalagmite as a input 
        input_vector(2)=yy(i,j);
        f(i,j) = stalagmite(input_vector);% calling the stalagmite function.
    end
end
num_cases=50% defining the number of itiration 
tic  %tic and toc command to find the study solving time 

for k=1:num_cases;%for loop for solving the each value for 50 itiration
    [input,fopt(k)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]); % bounding the upper limits and lower limits for GA search 
    xopt(k)=input(1)
    yopt(k)=input(2)% optimum falues of X and Y
end
surf(xx,yy,f);% ploting the 3D graph 
shading interp %remove mesh line from the graph
study_time=toc %record the study time 
figure(1) %figure 1
subplot(2,1,1)%creating subplot on figure  
hold on% hold the previous graph 
surfc(x,y,f)%creating the graph
shading interp 
plot3(xopt,yopt,fopt,'marker','o','markersize',5) %ploting the 3D graph
title('Bounded value')% title 
xlabel('X axis')% defining axis 
ylabel('Y axis')
zlabel('Globel maxima/Minima ')
subplot(2,1,2)
plot(fopt)% ploting 2d plot for number of itirations and findings of maxima 
xlabel('Iteration')
ylabel('Function Maximum')

Graphical results of the above code.

time taken by the study 52 sec

Conclusion for this study

limiting the search area provides more optimized solution

Genetic algoritm study 3

Reason for this Study 3

we need more optimized solution for which we are increasing the population size

clear all 
close all 
clc % clear all close all and clc to clear older data

x=linspace(0,0.6,150); %creating the search space for the ga in between it plot the graph
y=linspace(0,0.6,150);
 
[xx,yy]=meshgrid(x,y);% creating meshgrid for 3 dimensionl plot 
for i=1:length(xx); % nested for loop for processing each valu of xx and yy at a time
    for j=1:length(yy);
        input_vector(1)=xx(i,j); % defining the cordinates for ploting the graph and this cordinates will be used in stalagmite as a input 
        input_vector(2)=yy(i,j);
        f(i,j) = stalagmite(input_vector);% calling the stalagmite function.
    end
end
num_cases=50% defining the number of itiration 
tic  %tic and toc command to find the study solving time 
options=optimoptions('ga')
options=optimoptions(options,'populationsize',200);% increasing the population size 

tic
for k=1:num_cases;%for loop for solving the each value for 50 itiration
    [input,fopt(k)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);% syntax for increasing the size of population 
    xopt(k)=input(1)% optimum falues of X and Y
    yopt(k)=input(2)
end
surf(xx,yy,f);% ploting the 3D graph 
shading interp %remove mesh line from the graph
study_time=toc %record the study time 
figure(1) %figure 1
subplot(2,1,1)%creating subplot on figure  
hold on% hold the previous graph 
surfc(x,y,f)%creating the graph
shading interp 
plot3(xopt,yopt,fopt,'marker','o','markersize',5) %ploting the 3D graph
title('Bounded value')% title 
xlabel('X axis')% defining axis 
ylabel('Y axis')
zlabel('Globel maxima/Minima ')
subplot(2,1,2)
plot(fopt)% ploting 2d plot for number of itirations and findings of maxima 
xlabel('Iteration')
ylabel('Function Maximum')

Graphical results of study 3