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Challenge

Week 4.1 - Genetic Algorithm

BBabu Raj9 Jul 2022
MATLAB

AIM

Write a code in MATLAB to optimise the stalagmite function and find the global maxima of the function.

THEORY

  1. optimization

it is a field of study which deals with maximizing or minimizing of a function subject to different input parameters.

2.stalagmite mathematical model

it is a mathematical model used to find the local maxima or local minima using genetic algoritham .ga is explained in detail below.

3. Genetic algoritham

  • ga is an optimization technique which is used to solve non-linear ,non-differential optimization problem
  • it is based of mechanics of natural selection and natural genetics so called survival of fittest proposed by "Charles Darwin".
  • it starts with an initial generation of candidate solution that are tested against an objective function
  • subsequent generations evolve from 1st through selection ,cross over&mutation,
  • these process is explained in detail below

selection: In this, we are selecting the best performing bit strings from one generation to the next generation. i.e based on the Fitness.

Let, take an example

here we have 2 parents who have good performance so selecting them, based on the fitness.

Crossover: Due to they performed well, therefore taking common between them to create a child as shown below.

Mutation: Taking parent and mutate certain variable and create a child based on mutation. i.e in mutation, we are adding other qualities to the child.

syntax of ga

ga function is used to minimize a function hence inorder to maximize we inverse the function .

Syntax
x = ga(fun,nvars)
x = ga(fun,nvars,A,b)
x = ga(fun,nvars,A,b,Aeq,beq)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options)
x = ga(problem)
[x,fval] = ga(___)
[x,fval,exitflag,output] = ga(___)
[x,fval,exitflag,output,population,scores] = ga(___)

EXPLANATION

  • x = ga(fun,nvars,A,b) finds a local minimum x to fun, subject to the linear inequalities A*x ≤ b. ga evaluates the matrix product A*x as if x is transposed (A*x').
  • x = ga(fun,nvars,A,b,Aeq,beq) finds a local minimum x to fun, subject to the linear equalities Aeq*x = beq and A*x ≤ b. (Set A=[] and b=[] if no linear inequalities exist.) ga evaluates the matrix product Aeq*x as if x is transposed (Aeq*x').
  • x = ga(fun,nvars,A,b,Aeq,beq,lb,ub) defines 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.)
  • x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon) subjects the minimization to the constraints defined in nonlcon. The function nonlcon accepts x and returns vectors C and Ceq, representing the nonlinear inequalities and equalities respectively. ga minimizes the fun such that C(x) ≤ 0 and Ceq(x) = 0. (Set lb=[] and ub=[] if no bounds exist.)
  • x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options) minimizes with the default optimization parameters replaced by values in options. (Set nonlcon=[] if no nonlinear constraints exist.) Create options using optimoptions.
  • x = ga(fun,nvars,A,b,[],[],lb,ub,nonlcon,IntCon) or x = ga(fun,nvars,A,b,[],[],lb,ub,nonlcon,IntCon,options) requires that the variables listed in IntCon take integer values.
  • x = ga(problem) finds the minimum for problem, where problem is a structure.
  • [x,fval] = ga(___), for any previous input arguments, also returns fval, the value of the fitness function at x.
  • [x,fval,exitflag,output] = ga(___) also returns exitflag, an integer identifying the reason the algorithm terminated, and output, a structure that contains output from each generation and other information about the performance of the algorithm.
  • [x,fval,exitflag,output,population,scores] = ga(___) also returns a matrix population, whose rows are the final population, and a vector scores, the scores of the final population.

STALAGMITE FUNCTION 

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;
f2_x=exp(-4*log(2).*(((x-0.0667).^2)/0.64));
f2_y=exp(-4*log(2).*(((y-0.0667).^2)/0.64));
f=-(f1_x.*f1_y.*f2_x.*f2_y);
end

study-1: (statistical behavior)

clear all
close all
clc

% program for study 1 
% defining array 
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
[xx,yy]=meshgrid(x,y);
num_cases=150;

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
% 1st study _statistical behavior
tic
for i= 1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
study1_time =toc
fopt=-fopt;
figure(1)
subplot(2,1,1)
hold on 
surfc(x,y,-f)
shading interp
plot3(xopt,yopt,fopt,'marker','o','MarkerSize',6,'MarkerFaceColor','r')
title('unbounded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('iterations')
ylabel('function maximum')

EXPLANATION OF CODE

  • x,y are 1_D arrays created using linspace command
  • it is converted into 2-D using meshgrid command
  • num_cases=150 is a command used to run the ga code 150 times
  • then ga function is called and a no of variables is passed as input which is equal to 2 and storing x,yand f_opt value which function produced
  • tik and toc command is used to find the time it takes to measure study
  • new figure window is opened
  • subplot is created (creates multiple plots in same figure window)
  • 1st plot is surfc(x,y,f),plot 3 command is used for plotting x,y,ga_opt,which helps us to know how the algoritham perfoms
  • second subplot in which directly optimum value is plotted as a function of iterations is create

figure-1

RESULT

we get multiple results due to random sampling and maximum value varies from 0.2 to 0.8 . time taken for 1st study is 4.12735seconds .hence further optimization is requiered

study-2: (statistical behavior with upper and lower bounds)

[0,0]=upper and lower bound for x

[1,1]=upper and lower bound for y 

clear all
close all
clc

% program for study 2 
% defining array 
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
[xx,yy]=meshgrid(x,y);
num_cases=150;

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

%study 2 with upper and lower bounds
tic
for i=1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
fopt=-fopt;
study2_time=toc
figure(2)
subplot(2,1,1)
hold on 
surfc(x,y,-f)
shading interp
title('BOUNDED INPUTS')
xlabel('x values')
ylabel('y values')
plot3(xopt,yopt,fopt,'marker','o','MarkerSize',6,'MarkerFaceColor','r')
subplot(2,1,2)
plot(fopt)
xlabel('iterations')
ylabel('function maximum')

EXPLANATION

Here the only difference in program is along with ga function we set a upper bound and lower bound

figure -2

RESULT

now our results are more clear and the maximum range is between 0.7 to0.85.time taken for study=5.50825 seconds .still optimization is needed

STUDY-3 : (increasing no of iterations)

clear all
close all
clc

% program for study 1 
% defining array 
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
[xx,yy]=meshgrid(x,y);
num_cases=150;

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
% study 3 increasing GA iterations 
options=optimoptions('ga');
options=optimoptions(options,'PopulationSize',500);

tic
for i=1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6],[],[],options)
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
fopt=-fopt;
study3_time=toc
figure(3)
subplot(2,1,1)
hold on 
surfc(x,y,-f)
shading interp
title('increased iterations')
xlabel('x values')
ylabel('y values')
plot3(xopt,yopt,fopt,'Marker','o','MarkerSize',6,'MarkerFaceColor','r')
subplot(2,1,2)
plot(fopt)
xlabel('iterations')
ylabel('funtion maximum')

EXPLANATION

To optimize the results & getting more accuracy we increase the iteration in genetic algorithm.By default it takes 50 values .In this study we increase the population size becuase if population size is too small it will not calculate global maxima and minima instead it will calculate local max and min .By using options command we set population size to 500

figure-3

RESULT

now we get the most accurate result between 0.999999985-0.99999998.

time taken for study 3= 31.275 seconds

COMBINED STUDIES PROGRAM 

clear all
close all
clc

% program for study 1 
% defining array 
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
[xx,yy]=meshgrid(x,y);
num_cases=150;

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
% 1st study _statistical behavior
tic
for i= 1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
study1_time =toc;
fopt=-fopt;
figure(1)
subplot(2,1,1)
hold on 
surfc(x,y,-f)
shading interp
plot3(xopt,yopt,fopt,'marker','o','MarkerSize',6,'MarkerFaceColor','r')
title('unbounded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('iterations')
ylabel('function maximum')

%study 2 with upper and lower bounds
tic
for i=1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
fopt=-fopt;
study2_time=toc
figure(2)
subplot(2,1,1)
hold on 
surfc(x,y,-f)
shading interp
title('BOUNDED INPUTS')
xlabel('x values')
ylabel('y values')
plot3(xopt,yopt,fopt,'marker','o','MarkerSize',6,'MarkerFaceColor','r')
subplot(2,1,2)
plot(fopt)
xlabel('iterations')
ylabel('function maximum')

% study 3 increasing GA iterations 
options=optimoptions('ga');
options=optimoptions(options,'PopulationSize',500);

tic
for i=1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6],[],[],options)
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
fopt=-fopt;
study3_time=toc
figure(3)
subplot(2,1,1)
hold on 
surfc(x,y,-f)
shading interp
title('increased iterations')
xlabel('x values')
ylabel('y values')
plot3(xopt,yopt,fopt,'Marker','o','MarkerSize',6,'MarkerFaceColor','r')
subplot(2,1,2)
plot(fopt)
xlabel('iterations')
ylabel('funtion maximum')

max_value=fopt

RESULT

to get maximum value of the program we are using the function max_value=fopt and we get max value as 1.0000

CONCLUSION

1)Hence, we are able to optimise the stalagmite function and find the global maxima of the function.

2)The (ga) is only used for the minimum function therefore to Maximize the function we inversed the minimum function.

ERRORS FACED

ERROR 1: @ function not used infront of ga , error cleared by using @ga

apart from this no major errors occured .

REFERENCES 

 https://www.mathworks.com/discovery/genetic-algorithm.html
 https://en.wikipedia.org/wiki/Genetic_algorithm
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