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Challenge

Genetic Algorithm

RRahul Madaan29 Jun 2020
MATLAB

AIM:

To optimise the stalagmite function and find the global maxima of the function.

OBJECTIVES OF THE PROJECT:

Explaining the concept of genetic algorithm and also explain the syntax for ga in MATLAB.

Plot graphs for all 3 studies and for F maximum vs no. of iterations.

INTRODUCTION:

A Genetic Algorithm is a search experiential that is stimulated by Charles Darwin’s theory of natural evolution. This algorithm reproduces the process of natural selection where the appropriate individuals are selected for reproduction in order to produce offspring of the next generation.

The genetic algorithm is a technique for solving both constrained and unconstrained optimization problems that are based on natural selection, the process that drives biological evolution. The genetic algorithm constantly modifies the population size of individual solutions.

Phases in Genetic Algorithm-

  1. Initialization of Population (Coding)

The code generates its random population of defined size and from that population the code evaluates the fitness of the members of the population according to which it forms new population based on no. of iterations performed.

  1. Fitness Function

It is that function which we have to optimize and its optimization is done with the help of number of generations which have been formed by the code accordance with the fitness value of each member generated, these fitness values are used for the reproduction of the children.

  1. Selection

The idea of the selection phase is to select the fittest individuals and let them pass their genes to the next generation. The main goal of this phase is to find the region where the chances of getting the best solution are more.

Inspiration for this is from the survival of the fittest.

It should be a balance between exploration and exploitation of search space.

GA tries to move the genotype to higher fitness in the search space.

Too strong fitness selection bias can lead to sub-optimal solutions.

Too little fitness bias selection results in unfocused search.

  1. Reproduction

Generation of offspring happen in 2 ways:

Crossover

Mutation

  1. a) Crossover

Crossover is the most significant stage in the genetic algorithm. During crossover, a random point is selected while mating a pair of parents to generate offspring.

There are 3 major types of crossover.

Single Point Crossover: A point on both parents’ chromosomes is picked randomly, and designated a ‘crossover point’.

Two-Point Crossover: Two crossover points are picked randomly from the parent chromosomes.

Uniform Crossover: In a uniform crossover, typically, each bit is chosen from either parent with equal probability. The new offspring are added to the population.

  1. b) Mutation

In a few new offspring formed, some of their genes can be subjected to a mutation with a low random probability. This indicates that some of the bits in the bit chromosome can be flipped. Mutation happens to take care of diversity among the population and stop premature convergence.

FLOW CHART:

Syntax for GA:

[x,fval ,exitflag ,output ,population ,scores]= ga(func,nvars,A,b,Aeq,Beq,lb,ub,nonlcon,Intcon,options)

x = optimum solution

exitflag = reason ga stopped

output= information about optimization process

population=final population

scores= final scores

f=ga(fun,nvars,A,b,[],[],lb,ub,nonlcon,IntCon,options)

fun - fitness function.

nvars- number of variables,

A,b= Linear inequality constraints

Aeq,beq=Linear equality constraints

lb- Lower bound(limit),

ub- Upper bound(limit),

nonlcon : Use for nonlinear constraints

IntCon : to get solution in integer.

options- To set population size minimize with non-default options

BODY OF THE CONTENT:

  • FITNESS FNUCTION
  • The above equations are used in the stalagmite function to plot the graph in the main functions.
function [f] = 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;
 
            t1 = ((x-0.0667).^2)/(0.64);
            f2_x = exp(-4.*log(2)*t1);
 
            t2 = ((y-0.0667).^2)/(0.64);
            f2_y = exp(-4.*log(2)*t2);
 
            f = -(f1_x.*f2_x.*f1_y.*f2_y);
           
end
  • While using the function at the end as we are using the -ve sign so that the plot will be reversed as we have to find the maxima of the given function using ga.
  • MAIN BODY
    • Firstly, creating the mesh from the two vectors which will be acting as a search space to find the maxima of the function.
    • Creating the mesh of these two vectors using the ‘mesh’ command and defining the no. of iterations as num_cases to iterate the values of the optimum values of the function.\
    • Now calculating the values of fitness function at each point of the mesh grid formed.
    • Now creating its surface through ‘surfc’ command which will also provide the contour of the generated surface.
    • While using surfc applying the -ve sign to the function will generate the surface upside down which will be quite good to get the maxima of the function, as it will provide positive value for the maxima using the -ve sign.
clear
close all
clc

%% PROVIDING THE SEARCH SPACE FOR THE FUNCTION

    x=linspace(0,0.6,150);
    y=linspace(0,0.6,150);
    [X,Y]=meshgrid(x,y);
    
    num_cases=50;       %Number of itreation for which the function will evaluate the fitness


%% CREATING THE SURFACE PLOT OF STALAGEMITE FUNCTION

     for i=1:length(X)
         for j=1:length(Y)
             input_vector(1)=X(i,j);
             input_vector(2)=Y(i,j);
             f(i,j)=stalagmite(input_vector);   %values of function in given search space
         end
     end
 
        surfc(X,Y,-f);       %generating the surface
        shading interp      %To remove the lines produced by the mesh grid

The plot generated is shown below:

  1. STATISTICAL STUDY WITHOUT BOUNDATION
    • In this evaluating the optimum value of fitness function at each iteration and calculating it using the above-mentioned syntax in the intro part (which gives the inputs ie. Values of x and y at which the stalagmite function is minimum)
    • Then plotting it into the surf command and also providing the optimum values of the function that how it is varying.
    • Also using the tic and tok command to evaluate the time used for optimizing the function.
%% STATISTICAL BEHAVIOUR (UNBOUNDED)        
        
tic                             %calculating the study time through tic and toc command
for i=1:num_cases
    
        [inputs,fopt(i)]=ga(@stalagmite,2);     %calculating the optimized value of the function
        xopt(i)=inputs(1);                      %at each iteration
        yopt(i)=inputs(2);
    
end
study1_time=toc                  %displaying the time taken to evaluate
    
        % PLOTING THE GRAPH
        figure(1)

        subplot(2,1,1)

        surfc(X,Y,-f);      
        hold on
        shading interp
        plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
        xlabel('x');
        ylabel('y');
        grid on
        subplot(2,1,2)
        plot(-fopt);
        xlabel('iterations')
        ylabel('function min value')

RESULTS:

  • From the figure as shown above it is clear that the results are not that much accurate and giving a lot of variation in optimizing the function.

STATISTICAL STUDY WITH BOUNDATION

  • The difference over here is in providing the upper and the lower limit in the ga so that the search space is reduced with the upper and lower boundary
  • And then plotting it as similar to the above study.
%% STATISTICAL BEHAVIOUR (BOUNDED)

tic                                %calculating the study time through tic and toc command
 for j=1:num_cases
    
        [inputs,fopt(j)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]);   %calculating the optimized value of the function      
        xopt(j)=inputs(1);                                            %at each iteration with boundation
        yopt(j)=inputs(2);
    
 end
study2_time=toc
    
    %PLOTTING
    figure(2)
    
    subplot(2,1,1)
    hold on
    surfc(X,Y,-f)
    shading interp
    plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
    xlabel('x');
    ylabel('y');
    grid on
    subplot(2,1,2)
    plot(-fopt);
    xlabel('iterations')
    ylabel('function min value')

RESULTS

  • It is clearly seen from the above that after providing the boundary limits the variation in the optimum value of the function has reduced.

STATISTICAL STUDY WITH INCREASE IN SIZE OF POPULATION

  • In this study using the ‘optimoptions’ command to increase the size of population.
  • The benefit of that will be it will be working on the large set of numbers and which will be led to increase the possibilities of finding the better fitness of the function.
  • The drawback is that the time for evaluation will increase.
options=optimoptions('ga','populationsize',300)        %increaing the sixe of population by using
                                                        %optimoptions function
 
 tic                                                    
for k=1:num_cases
    
    [inputs,fopt(k)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);
     xopt(k)=inputs(1);
     yopt(k)=inputs(2);         
     
end
study3_time=toc
    
    %PLOTTING
    figure(3)
    
    subplot(2,1,1)
    hold on
    surfc(X,Y,-f)
    shading interp
    plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
    xlabel('x');
    ylabel('y');
    grid on
    subplot(2,1,2)
    plot(-fopt);
    xlabel('iterations')
    ylabel('function min value')

RESULTS

RESULTS

  • It is clearly seen in the above case that the value of the variation range has changed to a large extent.

NOW THE WHOLE PROGRAMA AT ONCE:-

clear
close all
clc

%% PROVIDING THE SEARCH SPACE FOR THE FUNCTION

    x=linspace(0,0.6,150);
    y=linspace(0,0.6,150);
    [X,Y]=meshgrid(x,y);
    
    num_cases=50;       %Number of itreation for which the function will evaluate the fitness


%% CREATING THE SURFACE PLOT OF STALAGEMITE FUNCTION

     for i=1:length(X)
         for j=1:length(Y)
             input_vector(1)=X(i,j);
             input_vector(2)=Y(i,j);
             f(i,j)=stalagmite(input_vector);   %values of function in given search space
         end
     end
 
        surfc(X,Y,-f);       %generating the surface
        shading interp      %To remove the lines produced by the mesh grid
       
        
       
%% STATISTICAL BEHAVIOUR (UNBOUNDED)        
        
tic                             %calculating the study time through tic and toc command
for i=1:num_cases
    
        [inputs,fopt(i)]=ga(@stalagmite,2);     %calculating the optimized value of the function
        xopt(i)=inputs(1);                      %at each iteration
        yopt(i)=inputs(2);
    
end
study1_time=toc                  %displaying the time taken to evaluate
    
        % PLOTING THE GRAPH
        figure(1)

        subplot(2,1,1)

        surfc(X,Y,-f);      
        hold on
        shading interp
        plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
        xlabel('x');
        ylabel('y');
        grid on
        subplot(2,1,2)
        plot(-fopt);
        xlabel('iterations')
        ylabel('function min value')
        
%% STATISTICAL BEHAVIOUR (BOUNDED)

tic                                %calculating the study time through tic and toc command
 for j=1:num_cases
    
        [inputs,fopt(j)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]);   %calculating the optimized value of the function      
        xopt(j)=inputs(1);                                            %at each iteration with boundation
        yopt(j)=inputs(2);
    
 end
study2_time=toc
    
    %PLOTTING
    figure(2)
    
    subplot(2,1,1)
    hold on
    surfc(X,Y,-f)
    shading interp
    plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
    xlabel('x');
    ylabel('y');
    grid on
    subplot(2,1,2)
    plot(-fopt);
    xlabel('iterations')
    ylabel('function min value')
   
%% MODIFYING THE NUMBER OF POPULATION 
    
 options=optimoptions('ga','populationsize',300)        %increaing the sixe of population by using
                                                        %optimoptions function
 
 tic                                                    
for k=1:num_cases
    
    [inputs,fopt(k)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);
     xopt(k)=inputs(1);
     yopt(k)=inputs(2);         
     
end
study3_time=toc
    
    %PLOTTING
    figure(3)
    
    subplot(2,1,1)
    hold on
    surfc(X,Y,-f)
    shading interp
    plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
    xlabel('x');
    ylabel('y');
    grid on
    subplot(2,1,2)
    plot(-fopt);
    xlabel('iterations')
    ylabel('function min value')

OUTPUT:

CONCLUSION:

ga is useful for optimizing the function but still the result are not accurate as the maximum value for the function lies at -1.

to overcome that using the given programme shown below 

clear
close all
clc

%% creating the mesh

x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
 [xx,yy]=meshgrid(x,y);
 
%% for plotting the surface of stalagmite

 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
 
%% evalutaing the function

    options=optimoptions('ga','PlotFcn',@gaplotbestf,'populationsize',200);
    [x,fval]=ga(@get_stalagmite,2,[],[],[],[],[0;0],[1;1],[],options)
    
        xopt=x(1);
        yopt=x(2);
        
            figure(1)

            surf(xx,yy,-f);
            shading interp
            hold on
            plot3(xopt,yopt,-fval,'ro','markersize',5,'markerfacecolor','r')

output

The above program will display the best value in the graph.

another way to find the best vaule is shown below.

  • NOTE:gaplotbestf, which plots the best and mean score of the population at every generation. Also setting gaplotstopping, which plots the percentage of stopping criteria satisfied.
clear
close all
clc

%% creating the mesh

x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
 [xx,yy]=meshgrid(x,y);
 
%% for plotting the surface of stalagmite

 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
 
 surfc(xx,yy,-f)
 shading interp
%% evalutaing the function

    options=optimoptions('ga','PlotFcn',{@gaplotbestf,@gaplotstopping},'populationsize',170);
    [x,fval]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],options)
    
        xopt=x(1);
        yopt=x(2);
        
            figure(1)
            hold on
            plot3(xopt,yopt,-fval,'ro','markersize',5,'markerfacecolor','r')

OUTPUT

It can be much more visualized that how the fitness is been calculated and best result is being dispalced even with the decrease in size of population.

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