Week 5 - Genetic Algorithm

• Fitness Function: This is the objective function to be minimized or maximized. It takes a vector of variables as input and returns a single scalar value representing the fitness or objective value.

• nvars: The number of variables in the problem.

• A, b, Aeq, beq: Matrices and vectors representing linear inequality and equality constraints.

• lb, ub: Vectors representing the lower and upper bounds for each variable.

• ConstraintFunction: (Optional) A function that provides nonlinear constraints in the form of c(x) <= 0 and ceq(x) = 0.

• options: (Optional) An options structure created using optimoptions to specify various settings like population size, number of generations, display options, etc.

• The optimoptions function is used to create an options structure that can be passed as an argument to various optimization solvers in MATLAB, including ga

The ga function returns the optimal solution vector x and its corresponding objective value fval. The GA searches for either the maximum or minimum value, depending on the nature of the problem.

clear 
close all
clc

% Define the stalagmite function
stalagmite_function = @(x) stalagmite([x(1); x(2)]);

% Input value in x and y
x_vals = linspace(0, 0.6, 150);
y_vals = linspace(0, 0.6, 150);

% number of iteration value
n = 50;

% Forming 2D array
[xx, yy] = meshgrid(x_vals, y_vals);
k = zeros(length(x_vals), length(y_vals));

% for loop for both x and y values  and K to calculate the stalagmite function
% values
for i = 1:length(x_vals)
    for j = 1:length(y_vals)
        input_vector = [xx(i, j); yy(i, j)];
        k(i, j) = stalagmite_function(input_vector);
    end
end

% Study 1 - Unbounded optimization
tic    % this is to start the clock for the study

% for loop for the ist unbounded study of the stalgamite
%The optimoptions function is used to create an options 
% structure that can be passed as an argument to various optimization solvers in MATLAB, including ga
for i = 1:n
    options = optimoptions('ga', 'Display', 'off');    % The option 'Display', 'off' means that the GA solver will not display any output during its execution 
    [results, fopt(i)] = ga(stalagmite_function, 2, [], [], [], [], [], [], [], options); % all of the syntax related information is explained in the report.
    xopt(i) = results(1);   %x-values after unbound optimization
    yopt(i) = results(2);   %y-values after unbound optimization
end
study_time1 = toc;   % end of the time for the study 1-unbound optimization

figure(1)                % creating the 1st figure
subplot(2, 1, 1)         % creating 1st subplot to show the animation 
surf(x_vals, y_vals, -k) % surf to represent the graph with x, y and k
shading interp           % to eliminate the mesh lines on the plot
hold on                   % hold on command wait for the next process
plot3(xopt, yopt, -fopt, 'marker', 'o')  % plot3 command to represent the way the unbound optimization works
subplot(2, 1, 2)           % creating 2nd subplot to show the graph
plot(-fopt)                % plot the values
xlabel('iteration')        % x label
ylabel('maximum value')    % y label

% Study 2 - Bounded optimization
tic        % this is to start the clock for the study

% for loop for the ist unbounded study of the stalgamite
%The optimoptions function is used to create an options 
% structure that can be passed as an argument to various optimization solvers in MATLAB, including ga
for i = 1:n
    options = optimoptions('ga', 'Display', 'off');
    lb = [0; 0];         %setting up the lower bound
    ub = [0.6; 0.6];     %setting up the upper bound
    [results, fopt(i)] = ga(stalagmite_function, 2, [], [], [], [], lb, ub, [], options);% all of the syntax related information is explained in the report.
    xopt(i) = results(1);   %x-values after bound optimization 
    yopt(i) = results(2);   %y-values after bound optimization 
end
study_time2 = toc;    % end of the time for the study 2-bound optimization

figure(2)                              % creating the 2nd figure
subplot(2, 1, 1)                       % creating 1st subplot to show the animation 
surf(x_vals, y_vals, -k)               % surf to represent the graph with x, y and k
shading interp                         % To eliminate the mesh lines on the plot
hold on                                % hold on command wait for the next process
plot3(xopt, yopt, -fopt, 'marker', 'o') % plot3 command to represent the way the bound optimization works
subplot(2, 1, 2)                        % creating 2nd subplot to show the graph
plot(-fopt)                             % plot the values
xlabel('iteration')                     % X LABEL
ylabel('maximum value')                 % y label

% Study 3 - Increased initial population
tic            % this is to start the clock for the study
% for loop for the ist unbounded study of the stalgamite
%The optimoptions function is used to create an options 
% structure that can be passed as an argument to various optimization
% solvers in MATLAB, including ga with increased population size

for i = 1:n
    options = optimoptions('ga', 'Display', 'off', 'PopulationSize', 300);
    [results, fopt(i)] = ga(stalagmite_function, 2, [], [], [], [], lb, ub, [], options);
    xopt(i) = results(1);      %x-values after bound optimization with increased population size
    yopt(i) = results(2);      %y-values after bound optimization with increased population size
end
study_time3 = toc;                     %% end of the time for the study 3-bound optimization with increased population size

figure(3)                              % creating the 3rd figure
subplot(2, 1, 1)                       % creating 1st subplot to show the animation 
surf(x_vals, y_vals, -k)               % surf to represent the graph with x, y and k
shading interp                         % To eliminate the mesh lines on the plot
hold on                                % hold on command wait for the next process
plot3(xopt, yopt, -fopt, 'marker', 'o') % plot3 command to represent the way the bound optimization works
subplot(2, 1, 2)                        % creating 2nd subplot to show the graph
plot(-fopt)                             % plot the values
xlabel('iteration')                     % X LABEL
ylabel('maximum value')                 % y label

%Stalgamite code
function A = stalagmite(input_vector)     % definition of stalgamite function
x=input_vector(1);                         % x-values from input
y=input_vector(2);                         % y-values from input

% Equations based on the x and y directions f1x and f1y and f2x and f2y
f1=(sin(5.1*pi*x+0.5)).^6;                   
f2=exp((-4*log(2)*((x-0.0667).^2))/0.64);
f3=(sin(5.1*pi*y+0.5)).^6;
f4=exp((-4*log(2))*((y-0.0667).^2)/0.64);
A =-(f1*f2*f3*f4);
end   % end of fuction                

Errors faced :

Err1:error in stalganmite_f in lines 3 and 28

resolution: changed the name of the function from the previous stalgamite_f to stalgamite_function

Err2: Invalid expression. when calling a function or indexing a variable, use parentheses. otherwise, check for mismatched delimiters.

resolution: added zeros in front of the expression the expression

Err3: error using optimoptions

resolution: display off instead of on, which explained in syntax

Err4:eror in ga

resolution: added one extra square bracket and semicolon

Explanation:

IN THE FIRST UNBOUND SIMULATION, WE DON'T HAVE ANY BOUNDS SETUP, SO THE SIMULATION WILL TAKE A NUMBER OF ITERATIONS TO SET DIFFERENT MAXIMUM VALUES .fOR THE NEXT BOUNDED SIMULATION WE HAVE THE LOWER BOUND AND UPPER BOUND BUT ONLY HAVE LESS NUMBER OF SAMPLE SIZE SO WE HAVE FEW MAXIMUM VALUES. AS WE INCREASE THE POPULATION SIZE FOR THE 3RD SIMULATION TO 300(TRIAL AAND ERROR),WE CAN SEE THE MAXIMUM VALUE IN THE SYSTEM.

REFERENCES;

1.https://www.mathworks.com/help/gads/ga.html

2.https://www.mathworks.com/discovery/genetic-algorithm.html

3.https://www.mathworks.com/help/gads/genetic-algorithm-options.html

4.https://www.mathworks.com/help/gads/options-and-outputs.html