Week 4.1 - Genetic Algorithm

• Selection rules select the individuals, called parents, that contribute to the population at the next generation. The selection is generally stochastic, and can depend on the individuals' scores.

• Crossover rules combine two parents to form children for the next generation.

• Mutation rules apply random changes to individual parents to form children.

Input Functions

Main Program

clear all
close all
clc

% Determining Search Space
x = linspace(0,0.6,150); % Values for X
y = linspace(0,0.6,150); % Values for Y
[xx,yy] = meshgrid(x,y); %Creating 2 D grid with x and y values

%calling the 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

%Iteration Number 1
num_cases_1 = 50; % Number of iterations
tic % Starting the time count

for i = 1:num_cases_1
    [inputs_1,F_optimum_1(i)] = ga(@stalagmite,2);
    x_optimum(i) = inputs_1(1);
    y_optimum(i) = inputs_1(2);
    optimum_function_1(i) = -F_optimum_1(i); % This gives us the maxima value
end

study_time_1 = toc; % Stopping the time count

figure(1)
subplot(2,1,1)
hold on
surfc(xx,yy,-F)
shading interp % To creat smooth curve surface
plot3(x_optimum,y_optimum,optimum_function_1,'Marker','o','MarkerFaceColor','r')
title('Unbounded Data')
subplot(2,1,2)
plot(optimum_function_1)
xlabel('Number of Iterations')
ylabel('F Maximum')


%Iteration Number 2
num_cases_2 = 100; % Number of iterations
tic % Starting the time count

for i = 1:num_cases_2
    % Setting upper and lower boundary
    [inputs_2,F_optimum_2(i)] = ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6]);
    x_optimum(i) = inputs_2(1);
    y_optimum(i) = inputs_2(2);
    optimum_function_2(i) = -F_optimum_2(i); % This gives us the maxima value
end

study_time_2 = toc; % Stopping the time count

figure(2)
subplot(2,1,1)
hold on
surfc(xx,yy,-F)
shading interp % To creat smooth curve surface
plot3(x_optimum,y_optimum,optimum_function_2,'Marker','o','MarkerFaceColor','r')
title('Bounded Data')
subplot(2,1,2)
plot(optimum_function_2)
xlabel('Number of Iterations')
ylabel('F Maximum')

%Iteration Number 3
num_cases_3 = 150; % Number of iterations
options = optimoptions('ga');
options = optimoptions(options,'PopulationSize',450);
tic % Starting the time count

for i = 1:num_cases_3
    % Setting upper and lower boundary
    % We use 'options' to define size of the population
    [inputs_3,F_optimum_3(i)] = ga(@stalagmite,2,[],[],[],[],[0,0],[0.6,0.6],[],options);
    x_optimum(i) = inputs_3(1);
    y_optimum(i) = inputs_3(2);
    optimum_function_3(i) = -F_optimum_3(i); % This gives us the maxima value
end

study_time_3 = toc; % Stopping the time count

figure(3)
subplot(2,1,1)
hold on
surfc(xx,yy,-F)
shading interp % To creat smooth curve surface
plot3(x_optimum,y_optimum,optimum_function_3,'Marker','o','MarkerFaceColor','r')
title('Bounded Data behaviour with increased population')
subplot(2,1,2)
plot(optimum_function_3)
xlabel('Number of Iterations')
ylabel('F Maximum')

% Global Maxima of every Iterations is the Maximum value in the
% corresponding 'F_optimum' array

The Stalagmite function Code

% Stalagmite Function that will be called in the maain function.
function [F] = stalagmite(input_vector)

x = input_vector(1);
y = input_vector(2);

F1 = (sin(5.1*pi*x + 0.5)).^6;
F2 = (sin(5.1*pi*y + 0.5)).^6;
F3 = exp((-4*log(2)*(x-0.0667).^2)/(0.64));
F4 = exp((-4*log(2)*(y-0.0667).^2)/(0.64));
F = -F1*F2*F3*F4;
end

Step By Step Explanations Of The Code.

Step 1-Clear all, close all, clc - to clear all the text form workspace,command window and to close all plots.

Step 2-We use 'linspace' command to define array of 150 equaly spaced numbers between 0 and 0.6 and is stored in the vaiables x and y.

Step 3-Now we creat a 2D mesh using 'meshgrid' command with x and y as inputs.

Step 4-Then we use 'for loop' to solve the 'stalagmite' function for every element in the 2D grid.

Step 5-Now we start the 1st iteration.

Step 6-For the 1st iteration number of casses denoted by 'num_cases_1' is 50 and at the start of the iteration we start the time count using command 'tic' to measure the time taken for the iterration.

Step 7-Then a for loop is used which iterate from 1 to number of element in 'num_cases_1'.

Step 8-Then we enter the 'ga' code in MATLAB which has the format '[inputs_1,F_optimum_1] = ga(fun,nvars). Here 'inputs' represent the values of parameter x and y for which the function has least value.

Step 9-The 'fun' and 'nvars' demots the values of parameter x and y for wchich the function has the least values and the number of variables respectively.

Step 10-Now we store the x and y coordinates of the minimum values of the fuction in variables 'x_optimum' and y_optimum' for all 50 iterations.

Step 11-Now we define the varable 'optimum_function' which will be the inverse of the minimum values, therefor 'optimum_function' will be the maxima values.

Step 12-Now we use the command 'toc' to stop the time count and store the time taken for the 1st iteration in the variable 'study_time_1'.

Step 13-Now we plot the figure for the 'unbounded data' by opening a new figure window and using 'subplot' command we can divide the current figure into an m-by-n grid and creates axes in the position specified by p.

Step 14-The 'surfc' command is used to plot contour with 'xx' and 'yy' values and 'optimum_function_1' since we need to represent the maxima of the function.

Step 15-We use 'shading interp' command to smoothen the function plot and to remove grid lines.

Step 16-Then we plot the 'optimum_function_1' on the same figure in the second row.

Step 17-Then we mention the 'xlabel' ,'ylabel' and the 'title'.

Step 18-Now we do the same procedure for 2nd iteration with 'num_case_2 = 100' (100 iterations) and has upper and lower limits (bounded).

Step 19-We use the 'ga' command with format '[inputs_2,F_optimum_2] = ga(fun,nvars,A,B,Aeq,Beq,lb,ub)'.

Step 20-A, B, Aeq, Beq are linear inequality constraints since all these are none we denot it as '[]'. 'lb' and 'ub' denots lower boundary and upper boundary.

Step 21-Now we plot the 'bounded data' similar to 'unbounded data' by opening another figure window.

Step 22-Now for the third iteration we use the same procedure with 'num_cases_3 = 150' (150 iterations) which has upper and lower bounds and also with increaed population size.

Step 23-Here we use the 'ga' command with format '[inputs_3,F_optimum_3] = ga(fun,nvars,A,B,Aeq,Beq,lb,ub,nonlcon or intcon,options)'.

Step 24-The 'intcon' and 'nonlcon' are integer variables and nonlinear constraints, since both has none values it is denoted as '[]'. Options denotes optimization options.

Step 25-This 'ga' is also bounded but with increase in population size, thus accuracy will be obtained.

step 26-After each iteration the time count is calculated.

Step 27-Now we plot the 'bounded data with incresed population' similar to the previous plots by opening another figure window.

Step 28-Now Global Maxima of every Iterations is the Maximum value in the corresponding 'F_optimum' array.

Step By Step Explanations Of The Stalagmite function Code.

Step 29-We define the function and mention its inputs.

Step 30-We have the equations for a stalagmite function. These equations are written and iputs are assigned.

Step 31-At last we compute the 'F' according to the input vectors.

Results

Plot -1 Unbounded Data

Plot - 2 Bounded Data

Plot - 3 Bounded data with increased population

Workspace Data

Syntax for 'ga' in MATLAB

F_min - Local min point.
@function - Calling stalagmite function.
nvars - Number of variables.
A, B, Aeq & Beq - Linear inequality constraints.
lb & ub - Lower and Upper boundary.
nonlcon - Nonlinear constraints.
intcon - Integer variables.
options - Used to perform increase in population calculation.

Errors while programming.

Error 1 - A typo error 'Y = linspace(0,0.6,150)' was typed instead of 'y = linspace(0,0.6,150)'
Solution - Change 'Y = linspace(0,0.6,150)' to 'y = linspace(0,0.6,150)'.

Error 2 - In line 27 'optimum_function' was typed instead of 'optimum_function_1'.
Solution - Change 'optimum_function' to 'optimum_function_1'.

Reference

1. https://in.mathworks.com/help/gads/how-the-genetic-algorithm-works.html
2. https://in.mathworks.com/help/gads/ga.html
3. https://en.wikipedia.org/wiki/Genetic_algorithm
4. https://www.geeksforgeeks.org/genetic-algorithms/
5. https://towardsdatascience.com/introduction-to-genetic-algorithms-including-example-code-e396e98d8bf3