Calculate the global maxima of a stalagmite function using Matlab

OBJECTIVE:

Write code in MATLAB to optimize the stalagmite function and find the global maxima of the function.

WHAT IS GENETIC ALGORITHM?

GA was proposed by CHALS DARWIN

in Matlab it\'s an inbuilt function

The genetic algorithm is a method for solving both constrained and unconstrained optimization problems that are based on natural selection, the process that drives biological evolution. The genetic algorithm repeatedly modifies a population of individual solutions. At each step, the genetic algorithm selects individuals at random from the current population to be parents and uses them to produce the children for the next generation.

You can apply the genetic algorithm to solve a variety of optimization problems that are not well suited for standard optimization algorithms

Such as discontinuous, nondifferentiable, stochastic, or highly nonlinear

WHY WE NEED TO USE GENETIC ALGORITHM :

Optimization is required in ic engine in aircraft mig airfoil for a proper lift at a different angle of attack and to minimize drag

Different flight condition required different drag value these should be minimum

It\'s a lengthy process

OPTIMIZATION: area of study of minimizing and maximize of function depends on input parameters

STALAGMITE FUNCTION: It is a mathematical model which helps us generate local minima and maxima

it gives a 3d model by which we can get the maxima and minima

To calculate the values of [x,y] and minimum value of the function as GA calculates the minimum values

we have a formula we need to convert it into a function

as stalagmite function

function akashmhaske = stalagmite(input_vector)
x=input_vector(1);
y=input_vector(2);
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)));
akashmhaske = (f1*f2*f3*f4);

end
   

as by staring the program by

clear all

close all

clc

define search space, we have defined values of \'x\' from 0 to 0.6 in a range of 150 values same for \'y\'

Mesh grid command used to create a gride of these values means 150*150 dimension

xx =150*150

yy =150*150

for calling the stalagmite function we need to use

FOR loop to define the values in a range of [xx] and [yy]

so that the loop should run for each value

as i = 1 : length(xx)

j =1:length(yy)

calling the function we get values of \'f\' in a 150*150 dimension

plot the graph of (xx, yy,f) ie inputs of xx, yy and function we got

we get a global maximum and local maximum plot but we only get the value of the minimum

clear all 
close all
clc

% defining our search space
x = linspace(0,0.6,150);
y = linspace(0,0.6,150)

% creating a 2dimension mesh
[xx, yy] = meshgrid(x, y);
% evaluating 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

surfc(xx,yy,f)
shading interp
xlabel(\'x value\')
ylabel(\'y value\')

[inputs,fval] = ga(@stalagmite,2)

MAIN PROGRAM FOR CALCULATING THE GA BUT ONLY GET MINIMUM VALUES

FIG.2 MESH GRID VALUES

FIG.3 GLOBAL AND LOCAL MAXIMA PLOT WITHOUT VALUES

FIG.4 VALUES OF INPUT VARIABLE AND FUNCTION VALUE MINIUM

TO GET THE GLOBAL MAXIMA WE NEED TO SOLVE GA BY CHECKING THREE CONDITIONS

Using the same function but this time minus sign when we multiply the f1,f2,f3,f4

define the range of x and y as GA takes random values and its a constrain the values of x and y to exceed beyond there range as Matlab doesn\'t understand it

function akashmhaske =get_stalagmite (xx,yy)

f1 = (sin(5.1*pi.*xx + 0.5)).^6;
f2 = exp(-4*log(2)*((xx-0.0667).^2/(0.64)));
f3 = (sin(5.1*pi.*yy + 0.5)).^6;
f4 = exp(-4*log(2)*((yy-0.0667).^2/(0.64)));
akashmhaske = -(f1.*f2.*f3.*f4);

end
   

giving num_cases = 50 means that the genetic algorithm will run for 50 times to give results

calling the function using get_stalagmite(xx,yy)

clear all 
close all 
clc


% defining our search space
x = linspace(0,0.6,150);
y = linspace(0,0.6,150)

% creating a 2dimension mesh
[xx, yy] = meshgrid(x, y);
num_cases = 50
f = get_stalagmite(xx,yy);

% STUDY 1 - SATISTICAL BEHAVIOR
tic
for i =1 : num_cases
        [inputs,fopt(i)] = ga(@stalagmite,2);
        xopt(i) = inputs(1);
        yopt(i) = inputs(2);     
end
study_time1 = toc
figure(01)
subplot(2,1,1)

surfc(-xx,-yy,-f)
hold on
shading interp
plot3(-xopt,-yopt,-fopt,\'marker\',\'o\',\'markersize\',5,\'markerfacecolor\',\'r\')
title(\'UNBOUNDED INPUTS\')
subplot(2 ,1 ,2)
plot(-fopt)

tic

% STUDY  2 - statisticaal behaviour - with upper and lower bonds
for i = 1:num_cases
    [inputs,fopt(i)] = ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]);
    xopt(i) = inputs(1);
    yopt(i) = inputs(2);
    
end
study_time2 = toc
figure(02)
subplot(2,1,1)

surfc(-xx,-yy,-f)
hold on
shading interp
plot3(-xopt,-yopt,-fopt,\'marker\',\'o\',\'markersize\',5,\'markerfacecolor\',\'r\')
title(\'BOUNDED INPUTS\')
subplot(2 ,1 ,2)
plot(-fopt)

% STUDY 3 increasing GA iterations 

options = optimoptions(\'ga\');
options = optimoptions(options,\'Populationsize\',370);


for i = 1:num_cases
    [inputs,fopt(i)] = ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);
    xopt(i) = inputs(1);
    yopt(i) = inputs(2);
    
end
study_time3 = toc
figure(03)
subplot(2,1,1)

surfc(-xx,-yy,-f)
hold on
shading interp
plot3(-xopt,-yopt,-fopt,\'marker\',\'o\',\'markersize\',5,\'markerfacecolor\',\'r\')
title(\'Bounded inputs increase the population size\')
subplot(2 ,1 ,2)
plot(-fopt)

FIRST STUDY:

• We define FOR loop for a range of num_cases and get output for x and y
• By getting the output we can compare them with the function and plotting them in a subplot
• Using of shading interp so that we do not get lines over the graph and we get a smooth plot of GA
• Calculate the time for how many secs the operation was carried out by using tic toc command

FIG.5 FIRST CASE PLOT OF DIFFERENT RANGE VALUES OF X, Y, FUNCTION

In the first study, we can see the maximum of the values is not been properly define we are unable to find it

lots of local maxima are been defined so to this plot is not proper we need to try another method

global maxima lie between the range of 0 to 1 but we can not see as there are lots of variation in subplot 2

SECOND STUDY:

• We define FOR loop for a range of num_cases and get output for x and y
• now we are using GA in and fixing the range of x and y like 0 to 1
• And the empty brackets for them to read the LINEAR EQUALITY AND LINER INEQUALITY
• LINEAR EQUALITY = less than equal to values
• LINER INEQUALITY = equal values
• By getting the output we can compare them with the function and plotting them in a subplot
• Using of shading interp so that we do not get lines over the graph and we get a smooth plot of GA
• Calculate the time for how many secs the operation was carried out by using tic toc command
• this takes much run time as compared to the first study
• values we get are close to the global maxima
• this problem can be solved using increasing the number of iteration

FIG.6 SECOND CASE PLOT OF DIFFERENT RANGE VALUES OF X, Y, FUNCTION

RESULT ARE IMPROVED

THIRD STUDY:

• We define the range of the population to 370 (population size )
• As the size is more the GA will take much time to give the result as compared to first and second
• We define FOR loop for a range of num_cases and get output for x and y
• Now we are using GA in and fixing the range of x and y like 0 to 1
• ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);
• The first four empty brackets for them to read the LINEAR EQUALITY AND LINER INEQUALITY
• LINEAR EQUALITY = less than equal to values
• LINER INEQUALITY = equal values
• [0;0],[1;1] is the range of x and y should be fixed
• another [],[] are for the linear equation with a degree and second bracket is for integers
• It takes thrice the time of the first case
• By getting the output we can compare them with the function and plotting them in a subplot
• Using of shading interp so that we do not get lines over the graph and we get a smooth plot of GA
• Calculate the time for how many secs the operation was carried out by using tic toc command
• This takes much run time as compared to the first study
• Values we get are close to the global maxima
• This problem can be solved using increasing the number of iteration
• We get the global maxima by this and time in which is finished is study time 112 sec

FIG.7 FIRST CASE PLOT OF DIFFERENT RANGE VALUES OF X, Y, FUNCTION

GOT THE GLOBAL MAXIMA NEGLECTING THE LOCAL MAXIMA

FIG.9 STUDY TIME OF OPERATION

CONCLUSION :

• The first case required less time but does not give an accurate result
• second also do the same
• As no of iteration is increased the GA shows the accurate value
• But we need to fix the range under which the input and output be defined