Genetic Algorithm

Genetic Algorithm (GA) is a search-based optimization technique based on the principles of Genetics and Natural Selection. It is frequently used to find optimal or near-optimal solutions to difficult problems which otherwise would take a lifetime to solve. It is frequently used to solve optimization problems, in research, and in machine learning.

A genetic algorithm is a search heuristic that is inspired by Charles Darwin’s theory of natural evolution. This algorithm reflects the process of natural selection where the fittest individuals are selected for reproduction in order to produce offspring of the next generation.

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

How the Genetic Algorithm Works

The following outline summarizes how the genetic algorithm works:

A stalagmite is a type of rock formation that rises from the floor of a cave due to the accumulation of material deposited on the floor from ceiling drippings. Stalagmites are typically composed of calcium carbonate, but may consist of lava, mud, peat, pitch, sand, sinter and amberat.

A stalagmite is an upward-growing mound of mineral deposits that have precipitated from water dripping onto the floor of a cave. Most stalagmites have rounded or flattened tips. There are many other types of mineral formations found in caves

Due to its shape we are using the stalagmite as the name for the program

STALAGMITE FUNCTION

A stalagmite function generally refers to the function with which we are generating values over a stalagmite kind of surface for finding out the local and global maxima and minima of that particular given function. 3. For this in Matlab we are using a method called 'Optimization'.

`f(x, y) = f1x*f1y*f2x*f2y`

where f1x, f1y, f2x, f2y

• `f1x = [sin(5.1*pi*x+0.5)]^6`

• `f1y = [sin(5.1*pi*y+0.5)]^6`

• `f2x=exp[-4ln(2)(x-0.0667)^2/(0.64)]`

• `f2y=exp[-4ln(2)(y-0.0667)^2/(0.64)]`

Programe for the stalagmite funtion in MATLAB

function [F] = stalagmitefuntion(input_vector)
 
x=input_vector(1);
y=input_vector(2);
    
    f1x = (sin((5.1*pi*x)+0.5))^6; 
    f1y = (sin((5.1*pi*y)+0.5))^6; 
    f2x = exp((-4*log(2)*(x-0.0667)^2)/0.64); 
    f2y = exp((-4*log(2)*(y-0.0667)^2)/0.64); 
    
    F = -(f1x*f1y*f2x*f2y);
end

• This is stalagmite function code, we have to call this function to the main programe

Code in MATLAB to optimise the stalagmite function and to find the global maxima of the function.

clear all
close all
clc
 
x=linspace(0,0.6,100);
y=linspace(0,0.6,100);
 
[xx,yy]=meshgrid(x,y);
 
for i=1:length(xx)
    for j=1:length(yy)
        
        inputs(1)=xx(i,j);
        inputs(2)=yy(i,j);
        F(i,j)=stalagmitefuntion(inputs);
    end
end
 
num=100;           % Itirations
tic
 
% Case1 : Statistical Behaviour
 
for i=1:num
   [inputs,Fopt(i)]=ga(@stalagmitefuntion,2);
    xout(i)=inputs(1);
    yout(i)=inputs(2);
end
 
 
 
casel_time=toc; 
 
figure(1)
subplot(2,1,1) 
surfc(xx,yy,-F) 
shading interp 
hold on
plot3(xout,yout,-Fopt,'marker','*','markersize',5,'markerfacecolor','r') 
title('Unbounded Inputs')
subplot(2,1,2) 
plot(-Fopt)
xlabel('Iterations')
ylabel('Function Maximum')
tic
 
%Case2 : Statistical Behaviour with upper bound & lower bound
 
for i=1:num
    [inputs,Fopt(i)]=ga(@stalagmitefuntion,2,[],[],[],[],[0;0],[0.6;0.6],[],[]);
    xout(i)=inputs(1);
    yout(i)=inputs(2);
end
 
case2_time=toc; 
 
figure(2)
subplot(2,1,1) 
surfc(x,y,-F)
shading interp 
hold on
plot3(xout,yout,-Fopt,'marker','*','markersize',5,'markerfacecolor','g')
title('Bounded Inputs')
subplot(2,1,2) 
plot(-Fopt)
xlabel('Iterations')
ylabel('Function Maximum')
tic
 
%Case3 : Increasing population
 
options=optimoptions('ga'); 
options=optimoptions(options,'populationsize',300);
 
for i=1:num
[inputs,Fopt(i)]=ga(@stalagmitefuntion,2,[],[],[],[],[0;0],[0.6;0.6],[],[],options);
xout(i)=inputs(1);
yout(i)=inputs(2);
end
 
case3_time=toc;
 
figure(3)
subplot(2,1,1)
surfc(x,y,-F) 
shading interp 
hold on
plot3(xout,yout,-Fopt,'marker','*','markersize',5,'markerfacecolor','b')
title('Bounded Inputs with Increased Population size')
subplot(2,1,2) 
plot(-Fopt)
xlabel('Iterations')
ylabel('Function Maximum')  

CODE EXPLINATION

• Creating a X and Y 1D vector 100 values b/w 0 to 0.6
• And by using the meshgrid we are creating the 2D vector where XX and YY
• By creating the for loop for I and J to the length of XX and YY
• inputs(1)=xx(i,j);
  inputs(2)=yy(i,j);
• And then we illbe calling the stalagmite functionand inserting the inputs in that
• Three cases

Case1 : Statistical Behaviour

Case2 : Statistical Behaviour with upper bound & lower bound

Case3 : Increasing population

Case1 : Statistical Behaviour

• To study the behaviour of the genetic algorithm without applying any condition to the optimisation process.

• Here we have to find out the global maxima so that we will be giving the negitive sign for the function

• Tic and toc command is used to get the study time taken for the ga to run

• For loop is used to run the ga for 100 times

• Through ga we will be calling the stalagmitefunction and giving the no of variables used in function is 2

• Outputs are

• xout(i)=inputs(1);
  yout(i)=inputs(2);

• And plotting graph using plot3 command

plot3(X,Y,Z) plots coordinates in 3-D space.

% Case1 : Statistical Behaviour
 
for i=1:num
   [inputs,Fopt(i)]=ga(@stalagmitefuntion,2);
    xout(i)=inputs(1);
    yout(i)=inputs(2);
end
 
 
 
casel_time=toc; 
 
figure(1)
subplot(2,1,1) 
surfc(xx,yy,-F) 
shading interp 
hold on
plot3(xout,yout,-Fopt,'marker','*','markersize',5,'markerfacecolor','r') 
title('Unbounded Inputs')
subplot(2,1,2) 
plot(-Fopt)
xlabel('Iterations')
ylabel('Function Maximum')
tic

Case2 : Statistical Behaviour with upper bound & lower bound

• ga is defined with function name ,no of variables and also boundation for the optimisation is set to 0 to 0.6

CODE FOR THE CASE 2

%Case2 : Statistical Behaviour with upper bound & lower bound
 
for i=1:num
    [inputs,Fopt(i)]=ga(@stalagmitefuntion,2,[],[],[],[],[0;0],[0.6;0.6],[],[]);
    xout(i)=inputs(1);
    yout(i)=inputs(2);
end
 
case2_time=toc; 
 
figure(2)
subplot(2,1,1) 
surfc(x,y,-F)
shading interp 
hold on
plot3(xout,yout,-Fopt,'marker','*','markersize',5,'markerfacecolor','g')
title('Bounded Inputs')
subplot(2,1,2) 
plot(-Fopt)
xlabel('Iterations')
ylabel('Function Maximum')
tic

• Result plots are shown below

• Time taken to run code is 10.5900 s
• Here we are getting the some of maximum values for the given inputs but we need global maxima

Case3 : Increasing population

• In this case we are going to change the inputs
• We are going to increase the population number by using the option command in the ga

• Increasing the population to 300 will makes the function to run longer and calculate with more values gives the optimum value

CODE FOR THE CASE 3

%Case3 : Increasing population
 
options=optimoptions('ga'); 
options=optimoptions(options,'populationsize',300);
 
for i=1:num
[inputs,Fopt(i)]=ga(@stalagmitefuntion,2,[],[],[],[],[0;0],[0.6;0.6],[],[],options);
xout(i)=inputs(1);
yout(i)=inputs(2);
end
 
case3_time=toc;
 
figure(3)
subplot(2,1,1)
surfc(x,y,-F) 
shading interp 
hold on
plot3(xout,yout,-Fopt,'marker','*','markersize',5,'markerfacecolor','b')
title('Bounded Inputs with Increased Population size')
subplot(2,1,2) 
plot(-Fopt)
xlabel('Iterations')
ylabel('Function Maximum')

• Result plots are shown below

• Time taken to run code is 46.8178 s
• Here we are getting the global maxima

CONCLUSION

we can see that the result of the optmisation is improved while changing the condition for the optmisation of the function like

• boundry conditions for the optimisation process
• increasing the population size

But while increasing the condition for optimisation process through ga the time required fro the optimisation also increased a lot.

so we can get the most optimum results form optimisation but the time required for the optimisation is also going to increase with it.

ERRORS FACED

• Always look whether we are giving correct sign or not one example is shown below

• Number of inputs should be equal in the ga command or else its going to throw a error as shown