Optimizing stalagmite function to find global maxima using genetic algorith

OBJECTIVE : The main objective of this project is to find global maxima in stalagmite function using solver \'Genetic Algorithm\'

Genetic Algorithm : Genetic algorithm solves smooth or nonsmooth optimization problems with any types of constraints, including integer constraints. It is a stochastic, population-based algorithm that searches randomly by mutation and crossover among population members. It\'s 5 phases are

1. Initial population

2. Fitness function

3. Selection

4. Crossover

5. Mutation

This project is divide in three studies :

STUDY 1 :

Finding local unconstrained maximum without defining bounds or population

Syntax : x = ga(fun,nvars) finds a local unconstrained minimum, x, to the objective function, fun. nvars is the dimension (number of design variables) of fun.

STUDY 2 :

Finding the solution by defining upper and lower bounds thereby reducing inaccuracy.

Syntax : 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.)

STUDY 3 :

Finding the solution by defining upper, lower bounds and population size thereby reducing inaccuracy.

Syntax : 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.

MATLAB CODE:

Following is the main code which finds solution in stalgmite function using GA in 3 cases.

% Input variables
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);

num_case = 50; % No of iteration for the ga function


% Creating An Array
[xx, yy] = meshgrid(x,y);

% For loop for passing the stalagmite function
for i=1:length(xx)
    for j=1:length(yy)
       input(1) = xx(i,j);
       input(2) = yy(i,j);
        f(i,j) = stalagmite_func(input);
    end
end
surfc(x,y,-f)
colorbar
xlabel(\'x Values\')
ylabel(\'y Values\')
zlabel(\'Function Values\')
title(\'Stalagmite Function\')
shading interp

[x_output,fval] = ga(@stalagmite_func,2);

% CASE STUDY 1

tic
for i = 1:num_case
    [inputs, f_optima(i)] = ga(@stalagmite_func,2);
    x_optima(i) = inputs(1);
    y_optima(i) = inputs(2);
end

case1 = toc;

figure(2)
subplot(2,1,1)
surfc(x,y,-f)
shading interp
xlabel(\'x values\')
ylabel(\'y values\')
zlabel(\'Function Values\')
hold on 
 
plot3(x_optima,y_optima,-f_optima,\'marker\',\'o\',\'markersize\',5,\'markerfacecolor\',\'r\')
title(\'Unbounded Input\')
subplot(2,1,2)
plot(-f_optima);
xlabel(\'No Of Iterations\')
ylabel(\'Maximum Value Of The Function\')

%CASE STUDY 2

tic
for i = 1:num_case
    [inputs, f_optima(i)] = ga(@stalagmite_func,2,[],[],[],[],[0;0],[0.5;0.5]);
    x_optima(i) = inputs(1);
    y_optima(i) = inputs(2);
end
case2 = toc;

figure(3)
subplot(2,1,1)
surfc(x,y,-f)
shading interp
xlabel(\'x Values\')
ylabel(\'y Values\')
zlabel(\'Function Values\')
hold on 

plot3(x_optima,y_optima,-f_optima,\'marker\',\'o\',\'markersize\',5,\'markerfacecolor\',\'r\')
title(\'Unbounded Input\')
subplot(2,1,2)
plot(-f_optima);
xlabel(\'No Of Iterations\')
ylabel(\'Maximum Value Of The Function\')

% CASE STUDY 3

options = optimoptions(\'ga\');   % Default values for the ga function
options = optimoptions(options, \'PopulationSize\', 200); % Setting the population size to 200

tic
for i=1:num_case
    [inputs,f_optima(i)] = ga(@stalagmite_func,2,[],[],[],[],[0;0],[0.5;0.5],[],[],options);
    x_optima(i) = inputs(1);
    y_optima(i) = inputs(2);
end

case3 = toc;
figure(4)
subplot(2,1,1)
surfc(x,y,-f)
shading interp
xlabel(\'x Values\')
ylabel(\'y Values\')
zlabel(\'Function Values\')
hold on 

plot3(x_optima,y_optima,-f_optima,\'marker\',\'o\',\'markersize\',5,\'markerfacecolor\',\'r\')
title(\'Unbounded Input\')
subplot(2,1,2)
plot(-f_optima);
xlabel(\'No Of Iterations\')
ylabel(\'Maximum Value Of The Function\')

Following is the equation to plot stalagmite function:

Following is the Matlab code for stalagmite function which incorporates above function:

function [f] = stalagmite_func(input)
X=input(1);
Y=input(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);  % -ve sign for finding global maximum
end

RESULTS :

This is plot of stalagmite function in which global maximum is to be found out.

STUDY 1:

In this case we can see the solution is random in nature since no bounds or population is defined.

STUDY 2 :

In this case we have defined upper and lower bounds and so the solution is lot closer to global maxima.

STUDY 3 :

In this case we have defined bounds as well as population size so the solver has accurately found global maximum. which is 1.00 and location is [x,y]=[0.0668,0.0668]

RESULT :

With this project we have found optimum solution in stalagmite function with the help of GA solver and were able to study statistical behaviour in various cases.