Week 4 - Genetic Algorithm

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.

Stalagmite Function Matlab Code

function formaximum = stalagmite(input_vector)
x=input_vector(1)
y=input_vector(2)

f1x = (sin(5.1*pi.*x+0.5))^6;
f1y = (sin(5.1*pi.*y+0.5))^6;
f2x = exp(-4*log(2).*(x-0.0667)^2 /0.64);
f2y = exp(-4*log(2).*(y-0.0667)^2 /0.64);
for_minimum = f1x.*f1y.*f2x.*f2y;
formaximum= -for_minimum 
end

Ga Optimisation codes

clear all 
close all
clc

x = linspace(0,0.6,150); % crating row matrix with 150 column
y = linspace(0,0.6,150); % crating row matrix with 150 column
[xx yy]= meshgrid(x,y); % creating 2D matrix
num_cases = 50; % number of times we want to run "ga"

% calculation of 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


% study-1 simply calling ga

% study-1 gives result of an unbounded system although we have given ga the
% boundary ie between the 0 and 0.6 but it does not takes those into
% account here 

% Now since the system is unbounde so the result we get will be different
% every time exexute the code

for i= 1:num_cases % i goes from 1 to 50
    % here inputs = the input at which the function is maximum
    
    % f_optimum is the optimum value of the function at those inputs
    
    %ga(@function name, number of variables in the function) so here we
    %have 2 variables these are x and y 
    
    [inputs, f_optimum(i)] = ga(@stalagmite, 2); % using ga
   % storing the input values as x and y optimum
   x_optimum(i)= inputs(1);
   y_optimum(i)= inputs(2);
end

figure(1)
% subplot is used to plot multiple plots ata time 
subplot(2,1,1)
%holding all the plots 
hold on
%generating the suface plot between all the inputs and the function 
 surfc(x,y,f)
 % shading interp removes the grid on the graph
shading interp
%ploting all the optimum values of x y and of the function for 50 times
%produced by ga

% they are shown by red dots and are connected though lines 
plot3( x_optimum, y_optimum, f_optimum,'marker','o','markersize',5,'markerfacecolor','r')
title('unbounded inputs')
subplot(2,1,2)
% plotting the optimum values to the iterations and seeing the deviation in
% the optimum value 
plot( f_optimum)
 xlabel(' iterations')
 ylabel(' function maximum')
 

% study-2 stastical behaviour whith forced upper and lower boundation
% study-2 gives result of an bounded system 


for i= 1:num_cases % i goes from 1 to 50
    % here inputs = the input at which the function is maximum
    
    % f_optimum is the optimum value of the function at those inputs
    
    %ga(@function name, number of variables in the function) so here we
    %have 2 variables these are x and y 
    
    %[0;0] is the lower bound value
    %[1;1] is the upper bound value
    
    [inputs, f_optimum(i)] = ga(@stalagmite, 2,[],[],[],[],[0;0],[1;1]); % using ga
   % storing the input values as x and y optimum
   x_optimum(i)= inputs(1);
   y_optimum(i)= inputs(2);
end

figure(2)
% subplot is used to plot multiple plots at a time 
subplot(2,1,1)
%holding all the plots 
hold on
%generating the suface plot between all the inputs and the function 
 surfc(x,y,f)
 % shading interp removes the grid on the graph
shading interp
%ploting all the optimum values of x y and of the function for 50 times
%produced by ga

% they are shown by red dots and are connected though lines 
plot3( x_optimum, y_optimum, f_optimum,'marker','o','markersize',5,'markerfacecolor','r')
title('Bounded inputs')
subplot(2,1,2)
% plotting the optimum values to the iterations and seeing the deviation in
% the optimum value 
plot( f_optimum)
 xlabel(' iterations')
 ylabel(' function maximum')


% study-3 stastical behaviour whith forced upper and lower boundation with
% more number of sample sapace or with more no of samples
% study-2 gives result of an bounded system 

% increasing the no of sampleas y increasing the populationsize of 170
options = optimoptions('ga')
options = optimoptions(options,'populationSize',170);


for i= 1:num_cases % i goes from 1 to 50
    % here inputs = the input at which the function is maximum
    
    % f_optimum is the optimum value of the function at those inputs
    
    %ga(@function name, number of variables in the function) so here we
    %have 2 variables these are x and y 
    
    %[0;0] is the lower bound value
    %[1;1] is the upper bound value
    % now searching from a large populatuonsize although 
    [inputs, f_optimum(i)] = ga(@stalagmite, 2,[],[],[],[],[0;0],[1;1],[],[],options); % using ga
   % storing the input values as x and y optimum
   x_optimum(i)= inputs(1);
   y_optimum(i)= inputs(2);
end

figure(3)
% subplot is used to plot multiple plots at a time 
subplot(2,1,1)
%holding all the plots 
hold on
%generating the suface plot between all the inputs and the function 
 surfc(x,y,f)
 % shading interp removes the grid on the graph
shading interp
%ploting all the optimum values of x y and of the function for 50 times
%produced by ga

% they are shown by red dots and are connected though lines 
plot3( x_optimum, y_optimum, f_optimum,'marker','o','markersize',5,'markerfacecolor','r')
title('Bounded inputs with more no of samples')
subplot(2,1,2)
% plotting the optimum values to the iterations and seeing the deviation in
% the optimum value 
plot( f_optimum)
 xlabel(' iterations')
 ylabel(' function maximum')

Results

Result-1 - Plots and results for unbounded system

Result-2 - Plots and results for Bounded System with lower and upper limit

Result-3 - Plots and results for unbounded system with lower and upper limit and a populationsize (sample space) of 170

Error and correction-1

The error came because of wrong syntax used it was supposed to be written as plot3 but I wrote plot(3) which made the syntax incorrect

Error and correction-2

Haven’t found any code for the get_stalagmite function so unless of making a new function I generated the stalagmite function in the codes itself

Error and correction-3

Syntaxt error as I wrote populationSize as poopilationsize out of excitement and corrected it afterwords

Refferences

https://in.mathworks.com/help/gads/ga.html?s_tid=mwa_osa_a#mw_5430547e-4111-4753-a80d-727f85b8f63e

https://www.youtube.com/watch?v=Q530u_g2CWE&list=PLYwpaL_SFmcDHUTN26NXKfjg6wFJKDO9R

https://skill-lync.com/challenges/question/SVhmM2VBYTRTREcyeEpHYUE4MUNFUT09OjpGWTZQb2E5ZmxTMW9xa1ZT/Z3BNUzRCdFkxdVFyTWlLZEx0OFpYZz09OjpKU2xIbWdsb3RuMENzNDdB/SW0yTUlPSWRjbWU3MjRDNXdnU0lCQT09OjpWVWViWExBdHVkYmRxc1Zu