CALCULATING THE GLOBEL MAXIMA BY USING GENETIC ALGORITHM

AIM:-

1.To find the globel maxima by using genetic algorithm

2.Need to write code in MATLAB to optimise the stalagmite function and find the global maxima of the function.

3.Expalination of genetic algorithm.

OBJECTIVES:-

1.Need to wright the stalagmite function to find the globel maxima.

2.Explination on genetic algorithm

THEORY:-

Genetic algorithmare commonly used to genarate high-quality solutions to optimization and search problems by relying on biologically inspired operators such as mutation,crossover and selection.

Jhon holland introduced genetic algorithms in 1960 based on the consept of Darwins's theory of evolution.

Optimization:-

Optimization is the selection of best element from the some set of available alternatives to find the minima & maxima of given problem.

What are the syntax we are genetic algorithm:-

1. [x favl]=ga (fun , nvars)

where

x=inputs

fval=function value

ga=genetic algorithm

fun=function name

nvar=number of variables

2.[x fval]=ga(fun,nvars,a,b,aeq,beq,lb,ub)

where

a,b= linear inequality constrains

aeq,beq=liner equality constrains

lb=lower bound

up=upper bound

3.[x fval]=ga(fun,nvars,a,b,[],[],lb,ub,nonicon,intcon,options)

where

nonicon=non-linear constraints

intcon=inter constraints

options=optimoptions

4.options=optimoptions(solver name)

solver name=ga

5.optins=optimoptions(solver name,name,value)

where

name=population size

value=population size value

Formula of function:-

(Star_fun) function program:-

Here we are creating the function according to the above show formula and to get the x,y variables ,thought behind this function is stalagmite

and We are calling the function with name star_fun.

function [f] = star_fun(input_vector)
x=input_vector(1);
y=input_vector(2);
s1x = (sin(5.1*pi*x+0.5))^6;
s1y = (sin(5.1*pi*y+0.5))^6;
s2x = exp(-4*log(2)*((x-0.0667)^2)/0.64);
s2y =  exp(-4*log(2)*((y-0.0667)^2)/0.64);
[f]=-s1x*s1y*s2x*s2y;
end
    

Stalagmite :-

Stalagmite is type of rock formation that rises from the floor of a cave due to the accumulation of material deposited on the floor from celling dripping.

Figure:-

Based on stalagmite function we are working to get global maxima

Main program :-

clear all
close all
clc

%Defining our surch space 

 x=linspace(0,0.6,150);
 y=linspace(0,0.6,150);
 num_cases=50;

% Creating 2Dimensional 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) = star_fun(input_vector);
    end
end
 
 surf(xx,yy,-f);

shading interp

tic
%study1-nstatical behaviour

for i=1:num_cases
    [inputs , fopt(i)]=ga(@star_fun,2);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end 
study1_time=toc
figure(1);
subplot(2,1,1);
hold on
surf(xx, yy, -f)
shading interp
 plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
title('un bounded input');
subplot(2,1,2);
plot(-fopt);
xlabel('interations');
ylabel('function minimum');
  
 tic
 %study2-nstatical behaviour

for i=1:num_cases
    [inputs, fopt(i)]=ga(@star_fun,2,[],[],[],[],[0,0],[0.7,0.7]);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end 
study2_time=toc
figure(2);
subplot(2,1,1);
hold on
surf(xx, yy, -f);
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('bounded input');
subplot(2,1,2);
plot(-fopt);
xlabel('interations');
ylabel('function minimum');


tic
%study3-nstatical behaviour
options=optimoptions('ga');
options=optimoptions(options,'populationsize',150)
for i=1:num_cases
    [inputs, fopt(i)]=ga(@star_fun,2,[],[],[],[],[0,0],[0.7,0.7],[],[],options);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end 
study3_time=toc
figure(3);
subplot(2,1,1);
hold on
surf(xx, yy, -f);
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
title('bounded input');
subplot(2,1,2);
plot(-fopt);
xlabel('interations');
ylabel('function minimum');

Figure(1):-

Output : without any values of optimum plots

Here in fig(1) it will show local maxima and local minimas

Figure(2):-

STUDY-1:STATISTICAL BEHAVIOUR HERE WE ARE NOT PROVIDING LOWER BOUNDS ,UPPER BOUNDS AND POLUTION SIZE

The above figure shows optimized maximum values after 50 iterations, for each iteration which are ploted in red dots are not bounded in specific area, so they are scattered all over the graph.

Study takes= 3.9341sec to complite the 50 itrations

Figure(3):-

STUDY-2: STATISTICAL BEHAVIOUR HER WE ARE PROVIDING LOWER BOUNDS AND UPPER BOUNDS BUT NOT PROVIDING POPULATION SIZE

The above figure shows optimized maximum values after 50 itrations,for each itration will be ploted in red are actually with in the area which we have specified,values which are not in the specified area are not consider.so we will get the best results for the globel maxima with high accurecy.

study takes=5.4808sec to complite the 50 itrations

Figure(4):-

STUDY-3: STATISTICAL BEHAVIOUR HERE WE WRE PROVIDING LOWER BOUNDS,UPPER BOUNDS AND POPULATION SIZE

Since the population size increassed the above figure shown global maxima

Study takes=13.8301sec to complite it 50 itrations of 150 population size

CONCLUSION:-

1.In 'MATLAB' we use a function solver is called 'ga',here 'ga' is used to solve and find the global minima's here we have to find 'maxima's so we add "-" negative sign to function values to get globel maxima

2.In this case we are taken no.of ITRATION'S as 50 (num_case=50) to find very accurate value of globel maxima