Week 4.1 - Genetic Algorithm

OBJECTIVE :

    Writing Matlab code the concept of genetic algorithm & optimize the stalagmite function and find the GLOBAL MAXIMA of the function.

GENETIC ALGORITHM :

  The genetic algorithm is amethod for solving both constrained and unconstrained optimization problems that is based on natural selection,the process that drives biological evolution.The genetic algorithm repeatedly modifies a population of individual solutions.At each step, the genetic algorithm selects individuals at random from the current population to be parents and uses them to produce the children for the next generation.Over successive generations, the population "evolves" towards an optimal solution.You can apply the genetic algorithm to solve a variety of optimization problems that are not well suited for standard optimization algorithms,including problems in which the objective function is discontinuous,nondifferentiable,stochastic or highly nonlinear.The genetic algorithem can address problems of mixed-integer programming,where some components are restricted to be integer-valued.

 

The genetic algorithm uses  three main types of rules at each step to create the next generation from the current population:

• Selection rules select the individuals, called parents,that contribute to the population at the next generation.
• crossover rule combine two parents to form childrean for the next generation.
• Mutation rules aplly random changes to individual parents to form children.

OPTIMIZATION :

Optimization is the process of making something better.In any process we have a set of inputs and set of outputs.

 

                                   

Optimization refers to finding the values of input such a way that we get the "best" output values.The defenition of the "best" varies from prpblem to problem,but in the mathematical terms,it refers to maximizing or minimizing one or more objective functions by varing the input parameters 

The set of all possible input values which the input can take make up the search space.In such a space, lies apoint or set of points  which gives the optimal solution. The aim of optimization to find tha point or set of point in the search space

GLOBAL MAXIMA:

A global maximum, also known as an absolute maximum the largest overall value of a set,function,etc.. over its entire range . It is impossible to construct an algorithm that will find global maximaum for an arbitary function.SEE also Global Minimum,Local Maxima,Maximum.

 

MAIN PROGRAM:

% Define search space
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);
num_cases = 50
% calculating 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);
        GA(i,j) = stalagmite(input_vector);
       
    end
end

% case 1
% study1-statical behaviour
tic


for i = 1:num_cases
[input, GA_opt(i)] = ga(@stalagmite,2);
xopt(i) = input(1);
yopt(i) = input(2);
end

study1_time = toc

%Plotting the graph

figure(1)
subplot(2,1,1)
hold on 
surfc(x,y,-GA)
shading interp
plot3(xopt,yopt,-GA_opt,'marker','o','markersize',5,'markerfacecolor','r')
title('Unbounded inputs')
subplot(2,1,2)
plot(-GA_opt)
xlabel('Iteration')
ylabel('Function Minium')

%% statistical behaviour - with upper & lower bounds
% tic_toc command for calculating the itration time
tic

for i = 1:num_cases
    [input, GA_opt(i)] = ga(@stalagmite,2,[],[],[],[],[0,0],[1,1]);
    xopt(i) = input(1);
    yopt(i) = input(2);
end
study2_time = toc

%plotting the Graph
figure(2)
subplot(2,1,1)
hold on 
surfc(x,y,-GA)
shading interp
plot3(xopt,yopt,-GA_opt,'marker','o','markersize',5,'markerfacecolor','r')
title('Bounded inputs')
subplot(2,1,2)
plot(-GA_opt)
xlabel('Iteration')
ylabel('Function Minium')

%% study-3
%calculating the optimum solution for 'ga' and save in options(variable)

options = optimoptions('ga');

%increasing the size of population

options = optimoptions(options,'PopulationSize',240);

%tic_toc command for calculating the itraction time

tic

for i = 1:num_cases
    [input, GA_opt(i)] = ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);
    xopt(i) = input(1);
    yopt(i) = input(2);
end
study3_time = toc

%plotting the Graph
figure(3)
subplot(2,1,1)
hold on 
surfc(x,y,-GA)
shading interp
plot3(xopt,yopt,-GA_opt,'marker','o','markersize',5,'markerfacecolor','r')
title('Bounded inputs')
subplot(2,1,2)
plot(-GA_opt)
xlabel('Iteration')
ylabel('Function Minium')

PROCEDURE:

step1: Assigning the input values range value of x & y is 0,0.6,150 values with the command of  linspace for both x & y

      2:mesh grid command used to create the grid size of these vale 150 * 150 dimension.

             xx =150*150

             yy =150*150

      3:For loop are used to define the value in a range of [xx],[yy] according to that its run the defined value.

      4:command with tic & toc used to calculate the simulation time

      5:using  shading interp so that we do not get lines over the graph and we get a smooth plot on it.

      6:seperate function should be used to calculate the stalagmite and its call to the main function. 

             formula to using in stalagimite function

 

                            

Stalagmite Function code:

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

term1 = pi*5.1*x;
term2 = pi*5.1*y;
term3 = ((x-0.0667)^2/0.64);
term4 = ((y-0.0667)^2/0.64);
term5 = (-4)*log(2);

xf1 = (sin(term1+0.5)).^6;
yf1 = (sin(term2+0.5)).^6;
xf2 = exp(term5*term3);
yf2 = exp(term5*term4);
GA = -(xf1*yf1*xf2*yf2);

end

Study1 : OUTPUT

STUDY1 time = 3.6284

From the above fig,it shows that its not clearly to define the maximum values due to a lot of minimum value it has been defined

STUDY2 : output

STUDY2 time = 4.0473

In study 2 shows that with the same procedure are followed to improve the result to find the global maxima with lower bonds variable are provided 

with the help of tic and toc command used to calculate the simulation time

The empty brackets are used to read the linear equality and in equality.

 

STUDY3 : OUTPUT

STUDY3 TIME = 12.2018

In study3 we have another constraint function as options as it helps us to increase the population to increase the size,Getting the local minima and maxima

With the help of tic and toc command used to calculate the simulation time 

 

ERRORS :

CONCLUSION:

From the above results we clearly see that optimizing the global maxima with various studies in that we need to get accurate value means we need to increase through the population options to get required output according to the value.

REFERENCE :

https://skill-lync.com/knowledgebase/genetic-algorithm-explained

https://www.tutorialspoint.com/genetic_algorithms/genetic_algorithms_introduction.htm

GOOGLE DRIVE :

https://drive.google.com/drive/folders/1D4Gdoktia7mQ_n2sWNPy55vkk1Z4TwFW?usp=sharing