Week 5 - Genetic Algorithm

The format for writing genetic algorithms in MATLAB is as follows:

1. x is the local minima point.
2. fun is the function to be called. (In our case, its the stalagmite function)
3. nvars indicates the no of variables used.
4. A and b are used to indicate any kind of inequalities that exist in the function.
5. Aeq and Beq also denote inequalities.
6. lb = lower bound.
7. ub = upper bound.
8. nonlcon gives two vectors representing inequalities and equalities.
9. options are used to perform additional operations like increasing population size. This is done with the help of "optimoptions".

clear all
close all
clc
x=linspace(0,0.6,150); % X value inputs
y= linspace(0,0.6,150);  % Y value inputs
% Even though search space is defined GA Algorithm donot run within this searchspace
[xx,yy]= meshgrid(x,y);

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

num_cases=50; % GA runs for 50 iterations
tic   % Study Time count start
for i=1:num_cases
 [inputs,f_opt(i)]= ga(@stalagmite,2); % GA function is used to run the stalagmite data and give the optimal solutions.
 x_opt(i)=inputs(1);
 y_opt(i)=inputs(2);
 opt_f(i)= -f_opt(i); %Ga gives us Minima value.Inverse f_opt is taken for Maxima value.

end
study1_time = toc   % Study Time count end 

subplot(2,1,1)
hold on
surfc(xx,yy,f) %Generation of stalagmite function.
shading interp  % To bring visually smooth curve surface without mesh lines
plot3(x_opt,y_opt,f_opt,'marker','o','markersize',4,'markerfacecolor','r')%plotting the optimum values with a red dot marker along the stalagmite 3D curves
title('Unbounded inputs')
subplot(2,1,2)
plot(opt_f) %plot of opt_f for every iterations
xlabel('Number of iterations ---->')
ylabel('Function Max -->')

2nd iteration:
Here the lower and upper bounds are defined so as to know the boundaries within which the global maxima lie. In other words, we are defining a definite space within which the global maxima are to be found.

clear all
close all
clc

x=linspace(0,0.6,150);  % X value inputs
y= linspace(0,0.6,150); % Y value inputs
[xx,yy]= meshgrid(x,y);

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); 
 %stalagmite function is called for every loop and output is stored in f
 end
end

num_cases=100; % GA runs for 100 iterations
tic         % Study Time start
for i=1:num_cases;
 % Objective function is stalagmite and the number of varaibles is 2
 % A,B,Aeq and Beq = [],as there is no Linear equalities and inequalities constraints in this function
 % lower boundary of varaibles x and y in ga alogorithmn is set as [0,0]
 % upper boundary of varaibles x and y in ga alogorithmn is set as [0.6,0.6]
 [inputs,f_opt(i)]= ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6]);
 x_opt(i)=inputs(1);
 y_opt(i)=inputs(2);
 opt_f(i)= -f_opt(i); %Ga gives us Minima value.Inverse f_opt is taken for Maxima value.
end
study2_time = toc  % Study time end
subplot(2,1,1)
hold on
surfc(xx,yy,f) %Generating the stalagmite function.
shading interp % To bring visually smooth curve surface without mesh lines
plot3(x_opt,y_opt,f_opt,'marker','o','markersize',4,'markerfacecolor','r')
title(' With Bounded inputs')
subplot(2,1,2)
plot(opt_f) %Plotting opt_f value for every iteration
xlabel('Number of iterations ---->')
ylabel('Function Max -->')

3rd iteration:
In this iteration, the population size is increased so as to converge to the Global maxima.

clear all
close all
clc
% x and y inputs
x=linspace(0,0.6,150);
y= linspace(0,0.6,150);
[xx,yy]= meshgrid(x,y);

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

num_cases=150; % GA runs for 150 iterations

options= optimoptions('ga');
options= optimoptions(options,'PopulationSize',450);
tic  % Study Time start
for i=1:num_cases
 % Objective function is stalagmite and the number of varaibles is 2
 % A,B,Aeq and Beq = [],as there is no Linear equalities and inequalities constraints in this function
 % lower boundary of varaibles x and y in ga alogorithmn is set as [0,0]
 % upper boundary of varaibles x and y in ga alogorithmn is set as [0.6,0.6]
 % population size is defined by using options in the algorithmn
 [inputs,f_opt(i)]= ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6],[],options);
 x_opt(i)=inputs(1);
 y_opt(i)=inputs(2);
 opt_f(i)= -f_opt(i); %Ga gives us Minima value.Inverse f_opt is taken for Maxima value.
end

Study3_time = toc   % Study Time end
subplot(2,1,1)
hold on
surfc(xx,yy,f) %Generating the stalagmite function.
shading interp % To bring visually smooth curve surface without mesh lines
plot3(x_opt,y_opt,f_opt,'marker','o','markersize',5,'markerfacecolor','r')
title('Bounded inputs with increased population size')
subplot(2,1,2)
plot(opt_f) %Plotting opt_f value for every iteration
xlabel('Number of iterations ---->')
ylabel('Function Max -->')
 

ERRORS while programming:

2)

While defining the inputs mentioned different sizes of input vectors x and y. The error is basically due to the mesh grid command where in the two vectors cannot be combined due to their varying sizes.

Apart from this some typos and errors.

This error was because of not defining the stalagmite.m file before running the code. This was rectified by defining and saving the stalagmite.m function file.

RESULTS & CONCLUSION:

The following are the results obtained on running the GA.

1st iteration: (Iterations = 50)

In the above image, it can be seen that since the inputs are boundless GA algorithm has mapped local maxima points outside our stalagmite function. Also, we cannot infer a particular value of global maxima from the above graph, and it is hence pointless.