STALAGMITE FUNCTION - OPTIMIZATION AND GLOBAL MAXIMA

clear all
close all
clc
 a = linspace(0,0.8,150);
 b = linspace(0,0.8,150);
 %creating input matrix
 [aa,bb] = meshgrid(a,b);
 
%obtaining the output from the function
for i = 1: length(a)
    for j = 1:length(b)
        inputs(1) = aa(i,j);
        inputs(2) = bb(i,j);
        P(i,j) = STALAGMITE(inputs);
        %P(i.j) is a matrix of output from the STALAGMITE function
    end
end
figure(1)
surfc(aa,bb,-P)
shading interp
title(\'FUNCTION PLOT\')
colorbar
xlabel(\'X-Axis\')
ylabel(\'Y-Axis\')
zlabel(\'Z-Axis\')

%fixing total number of cases.
nc = 50;
%case 1 - unbounded condition
for i = 1: length(a)
    for j = 1:length(b)
        inputs(1) = aa(i,j);
        inputs(2) = bb(i,j);
        P(i,j) = STALAGMITE(inputs);
        %P(i.j) is a matrix of output from the STALAGMITE function
    end
end
tic
for k = 1:nc
    [inputsS1, fnS1(k)] = ga(@STALAGMITE,2);
    xnS1(k) = inputsS1(1);
    ynS1(k) = inputsS1(2);
end
Runtime_unbound = toc;
%ploting the surface plot and marking the maxima
figure(2)
subplot(2,1,1)
surfc(aa,bb,-P)
%-ve for P since ga gives the minima and hence inverting for maxima.
shading interp
colorbar
title(\' UN-BOUND OR UN-CONSTRAINED\')
hold on
plot3(xnS1,ynS1,-fnS1,\'o-\',\'Markerfacecolor\',\'r\',\'Markersize\',4)
%using -fnS1 since the we have to plot the maxima points
xlabel(\'X-Axis\')
ylabel(\'Y-Axis\')
zlabel(\'Z-Axis\')
subplot(2,1,2)
plot(-fnS1)
%using -fnS1 since the we have to plot the fitness points
xlabel(\'No. of iterations\')
ylabel(\'Function MAX\')

%CASE 2 x = ga(fun,2,A,b,Aeq,beq,lb,ub) - providing boundaries
for i = 1: length(a)
    for j = 1:length(b)
        inputs(1) = aa(i,j);
        inputs(2) = bb(i,j);
        P(i,j) = STALAGMITE(inputs);
        %(i.j) is a matrix of output from the STALAGMITE function
    end
end
tic
for o = 1:nc
    [inputsS2, fnS2(o)] = ga(@STALAGMITE,2,[],[],[],[],[0;0],[0.4;0.4]);
    xnS2(o) = inputsS2(1);
    ynS2(o) = inputsS2(2);
end
Runtime_bound = toc;

%ploting the surface plot and marking the maxima
figure(3)
subplot(2,1,1)
surfc(aa,bb,-P)
%-ve for P since ga gives the minima and hence inverting for maxima.
shading interp
colorbar
title(\' BOUND OR CONSTRAINED\')
hold on
plot3(xnS2,ynS2,-fnS2,\'o-\',\'Markerfacecolor\',\'r\',\'Markersize\',5)
xlabel(\'X-Axis\')
ylabel(\'Y-Axis\')
zlabel(\'Z-Axis\')
subplot(2,1,2)
plot(-fnS2)
%using -fnS2 since the we have to plot the fitness points
xlabel(\'No. of iterations\')
ylabel(\'Function MAX\')

%CASE 3 Bounded with fixed Population
for i = 1: length(a)
    for j = 1:length(b)
        inputs(1) = aa(i,j);
        inputs(2) = bb(i,j);
        P(i,j) = STALAGMITE(inputs);
        %(i.j) is a matrix of output from the STALAGMITE function
    end
end

tic
options = optimoptions(\'ga\')
options = optimoptions(options,\'populationsize\',250)
for o = 1:nc
    [inputsS3, fnS3(o)] = ga(@STALAGMITE,2,[],[],[],[],[0;0],[0.4;0.4],[],options);
    xnS3(o) = inputsS3(1);
    ynS3(o) = inputsS3(2);
end
Runtime_pop_bound = toc;

%ploting the surface plot and marking the maxima
figure(4)
subplot(2,1,1)
surfc(aa,bb,-P)
%-ve for P since ga gives the minima and hence inverting for maxima.
shading interp
colorbar
title(\' BOUND OR CONSTRAINED with Population\')
hold on
plot3(xnS3,ynS3,-fnS3,\'o-\',\'Markerfacecolor\',\'r\',\'Markersize\',6)
xlabel(\'X-Axis\')
ylabel(\'Y-Axis\')
zlabel(\'Z-Axis\')
subplot(2,1,2)
plot(-fnS3)
%using -fnS3 since the we have to plot the fitness points
xlabel(\'No. of iterations\')
ylabel(\'Function MAX\')

type = [{\'unbound\';\'Bounded\';\'Bounded with options\'}];
time = [Runtime_unbound;Runtime_bound;Runtime_pop_bound];
RESULTS = table(type,time,\'VariableNames\',{\'STUDY TYPE\',\'STUDY TIME\'})

2) Function Definition

%Defining the Function
function [OP] = STALAGMITE(inputs)
  %defining x and y
 x = inputs(1);
 y = inputs(2);
  
f1x = (sin(((5.1*pi)*x)+(0.5)))^6;
f2x = exp((-4*log(2))*(((x - 0.0667)^2)/0.64));

f1y = (sin(((5.1*pi)*y)+(0.5)))^6;
f2y = exp((-4*log(2))*(((y - 0.0667)^2)/0.64));

%providing the output from the function
OP = -1*f1x*f2x*f1y*f2y;
 %-1 is multiplied to maximise the inputs or to obtain a max. value.
end

Code Explanation:

Line 4 - 7: Input array and matrix is defined and generated.

Line 10 - 17: using for loop to obtain output of the function for each value of the input matrix

Line 18 - 25 : Creating the plot of the given function using surfc command, -P is taken to plot the maximum values.

Line 28 : Fixing the numbe of iterstions required.

Line 30 - 37 : using for loop to obtain output of the function for each value of the input matrix

Line 39 - 43 : using genetic algorithm for the function to get maximised output. also tic - toc command used to find out the time taken for the current study.

Line 46 -53 : Creating the plot of the given function and condition using surfc command, -P is taken to plot the maximum values.

Line 54 - 63: plotting the points at which the function is having the maximum ( -fnS1 used to plot the maximum value, where fnS1 contains the minimum value)

Line 65 - 73 : using for loop to obtain output of the function for each value of the input matrix

Line 75 to 80 :using genetic algorithm for the function with boundaries to get maximised output. also tic - toc command used to find out the time taken for the current study.

Line 83 - 90 : Creating the plot of the given function and condition using surfc command, -P is taken to plot the maximum values.

Line 91 - 99 :plotting the points at which the function is having the maximum ( -fnS2 used to plot the maximum value, where fnS2 contains the minimum value)

Line 101 - 109 :using for loop to obtain output of the function for each value of the input matrix

Line 112-113 : defining options to get the global maxima value( defining the population size)

Line 114 - 119 :using genetic algorithm for the function with boundaries  and  population size to get maximised output. also tic - toc command used to find out the time taken for the current study.

Line 122 -129 : Creating the plot of the given function and condition using surfc command, -P is taken to plot the maximum values.

Line 130 - 139: plotting the points at which the function is having the maximum ( -fnS3 used to plot the maximum value, where fnS3 contains the minimum value)

Line 139 - 141: Taking the time taken to a table to display as output.

Function description:

initially x and y are defined as inputs to the functions respectively.

Then the definition of function is done.

The result of the function is assigned to a variable. The negative sign is required since Genetic algorithm minimises a function , so the negative of the function helps in creating the maximum values.

Syntax For GA :

Case 1

x = ga(fun,nvars)

produces minimum x for function \'fun\', having \'nvars\' number of design variables.

CASE 2

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub)

In this case, lower and upper boundary is specified where the solution lies in between the boundaries. Here since linear inequalities are not considered, replace A,b, Aeq, beq by [].

Case 3

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options)

\'nonlcon\' represents the non linear inequalities which is not considered,hence replaced by [] in the syntax. The options variable is used for optimisation within a specified size. options can be defined using optimoptions command.

RESULT:

1) Function Plot

2) Case 1 - UN-BOUNDED Condition

3) Case 2 - Function with Lower and Upper Boundaries

4) Case 3 - Function with Boundaries and Specified population size

5) Time To Study

Errors Faced:

1) Error : x and y value not defined for the function

    Correction: x and y defined within the function

    Screen Shot: 

2) Error : x and y value wrongly defined for the function

    Correction: x and y defined as per need of the function

    Screen Shot: 

3) Error : wrong description of command population size

    Correction: \'size\' changed to \'populationsize\'

    Screen Shot: 

4) Error : synatx error while using the ga command with boundaries and options

    Correction: missing value as per syntax was provided

    Screen Shot: 

CONCLUSION:

The maxima for the function was obtained by inversing the function output using Genetic Algorithm operation. Through the process of selection, cross-over and mutation, the Global Maxima for the function is produced. Time for each study reveals the amount of processing required for obtaining the Global Maxima starting from a set of Local Maximas.

REFERENCES:

https://link.springer.com/chapter/10.1007/978-3-540-39930-8_2

https://www.neuraldesigner.com/blog/genetic_algorithms_for_feature_selection

https://in.mathworks.com/discovery/genetic-algorithm.html