Week 4.1 - Genetic Algorithm
AIM
To optimise the stalagmite function and find the global maxima of the function using Matlab.
OBJECTIVES OF THE PROJECT
- Explaining the concept of genetic algorithm and also explain the syntax for ga in MATLAB.
- Plot graphs for all 3 studies and for F maximum vs No of iterations.
INTRODUCTION
A Genetic Algorithm is a search experimental that is stimulated by Charles Darwin's theory of natural evolution. This Algorithm reproduces the process of natural selection where the appropirate individuals are selected for reproduction in order to produce offspring of the next generation.
The Genetic algorithm is a technique for solving both constrained and unconstrained optimization problems that are based on natural selection,the process that drives biological evolution.The Genetic algorithm constantly modifies the population size of individual solutions.
PHASES IN GENETIC ALGORITHM
1.Initialization of population (coding)
The code generates its Random population of defined size and from that population the code evaluates the fitness of the members of the population according to which it forms new population based on Number of iterations performed.
2.Fitness function
It is that function which we have to optimize and its optimization is done with the help of number of generations which have been formed by the code accordance with the fitness values of each member generated, these fitness values are used for the reproduction of the children.
3.Selection
The Idea of the selection phase is to select the fittest individuals and let them pass their genes to the next generation.The main goal of this phase is to find the region where the chances of getting the best solution are more.
- Inspiration for this is from the survival of the fittest.
- It should be a balance between exploration and exploitation of search space.
- GA tries to move the genotype to higher fitness in the search space.
- Too strong fitness selection bias can lead to sub-optimal solutions.
- Too little fitness bias selection results in unfocused search.
REPRODUCTION
Generation of offspring happens in 2 ways:
- CROSSOVER
- MUTATION
CROSSOVER
Crossover is the most significant stage in the genetic algorithm.During crossover,a random point is selected while mating a pair of parents to generate offspring.
There are 3 major types of crossover .
Single Point Crossover : A point on both parents chromosomes is picked randomly, and designated a 'crossover point'.
Two point Crossover : Two crossover points are picked randomly from the parent chromosomes.
Uniform Crossover : In a uniform crossover,typically each bit is chosen from either parent with equal probability.The new offspring are added to the population.
MUTATION
In a few new offspring formed ,some of their genes can be subjected to a mutation with a low random probability.This indicates that some of the bits in the bit chromosome can be flipped.Mutation happens to take care of diversity among the population and stop premature convergence.
FLOW CHART
SYNTAX FOR GA:
[x,fval ,exitflag ,output ,population ,scores]= ga(func,nvars,A,b,Aeq,Beq,lb,ub,nonlcon,Intcon,options)
x = optimum solution
exitflag = reason ga stopped
output = information about optimization process
population = final population
scores = final scores
f=ga(func,nvars,A,b,Aeq,Beq,lb,ub,nonlcon,Intcon,options)
func = fitness function
nvars = number of variables
A,b = linear inequality constraints
Aeq,Beq = linear equality constraints
lb = lower bound(limit)
ub = upper bound(limit)
nonlcon = use for nonlinear constraints
Intcon = to get soution in integer
options = to set population size minimum with non-default options
BODY OF THE CONTENT:
Fitness Function :
- The above Equations are used in the stalagmite function to plot the graph in the main functions.
function [f] = stalagmite(input_vector)
x = input_vector(1);
y = input_vector(2);
f1_x = (sin(5.1*pi*x + 0.5)).^6;
f1_y = (sin(5.1*pi*y + 0.5)).^6;
t1 = ((x-0.0667).^2)/(0.64);
f2_x = exp(-4.*log(2)*t1);
t2 = ((y-0.0667).^2)/(0.64);
f2_y = exp(-4.*log(2)*t2);
f = -(f1_x.*f2_x.*f1_y.*f2_y);
end
- While using the function at the end as we are using the negative sign, So that the plot will be reversed as we have to find the maxima of the given function using ga.
MAIN BODY
- Firstly, creating the mesh from the two vectors which will be acting as a search space to find the maxima of the function.
- Creating the mesh of these two vectors using the 'mesh' command and defining the no of iterations as num_cases to iterate the values of the optimum values of the function.
- Now calculating the values of fitness function at each point of the mesh grid formed.
- Now creating its surface through 'surfc' command which will also provide the contour of thr generated surface.
- While using surfc applying the negative sign to the function will generate the surface upside down which will be quite good to get the maxima of the function, as it will provide positive value for the maxima using the negative sign.
close all
clear all
clc
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);
[x,y] = meshgrid(x,y);
num_cases = 50; %Number of iteration for which the function will evaluate the fitness
% Creating the surface plot of stalagmite function
for i = 1:length(x)
for j = 1:length(y)
input_vector(1) = x(i,j);
input_vector(2) = y(i,j);
f(i,j) = stalagmite(input_vector); %Values of function in given search space
end
end
surfc(x,y,-f); %Generating the surface
shading interp % To remove the lines produced by the meshgrid The Plot generated is shown below:
STATISTICAL BEHAVIOUR WITHOUT BOUNDATION
- In this evaluating the optimum value of fitness function at each iteration and calculating it using the above mentioned syntax in the intro part.
- Then plotting it into the surf command and also providing the optimum values of the function that how it is varying.
- Also using the tic and toc command to evaluate the time used for optimizing the function.
% Statistical behaviour (Unbounded)
tic % Calculating the study time using the command tic and toc
for i = 1:num_cases
[inputs,fopt(i)] = ga(@stalagmite,2); %Calculating the optimized value of the function
xopt(i) = inputs(1); % at each iteration
yopt(i) = inputs(2);
end
study1_time = toc %Displaying the time taken to evaluate
% Plotting the graph
figure(1)
subplot(2,1,1)
surfc (x,y,-f);
hold on
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
xlabel('x');
ylabel('y');
grid on
subplot(2,1,2)
plot(-fopt);
xlabel('iterations')
ylabel('function min value')RESULTS
- Find the figure as shown above,it is clear that the results are not that much accurate and giving a lot of variation in optimizing the function.
STATISTICAL STUDY WITH BOUNDATION
- The difference over here is in providing the upper and the lower limit in the ga so that the search space is reduced with the upper and lower boundary.
- And then plotting it as similar to the above study.
% Statistical Behaviour (Bounded)
tic % Calculating the study time through tic and toc command
for j = 1:num_cases
[inputs,fopt(j)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]); % Calculating the optimized value of the function
xopt(j)=inputs(1); % At each iteration with Boundation
yopt(j)=inputs(2);
end
study2_time=toc
% Plotting
figure(2)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
xlabel('x');
ylabel('y');
grid on
subplot(2,1,2)
plot(-fopt);
xlabel('iterations')
ylabel('function min value')RESULTS
- It is clearly seen from the above that after providing the boundary limits the variation in the optimum value of the function has reduced.
STATISTICAL STUDY WITH INCREASE IN SIZE OF POPULATION
- In this study using the 'optimoptions' command to increase the size of population.
- The benefit of that will be working on the large set of numbers and which will lead to increase the possibilities of finding the better fitness of the function.
- The drawback is that the time for evaluation will increase.
options = optimoptions('ga','populationsize',300) %Increasing the size of population by using the optimoptions function
tic
for k = 1:num_cases
[inputs,fopt(k)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);
xopt(k)=inputs(1);
yopt(k)=inputs(2);
end
study3_time = toc
% Plotting
figure(3)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
xlabel('x');
ylabel('y');
grid on
subplot(2,1,2)
plot(-fopt);
xlabel('iterations')
ylabel('function min value')RESULTS
- It is clearly seen in the above case that the value of the variation range has changed to a large extent.
WHOLE PROGRAM AT ONCE
close all
clear all
clc
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);
[x,y] = meshgrid(x,y);
num_cases = 50; %Number of iteration for which the function will evaluate the fitness
% Creating the surface plot of stalagmite function
for i = 1:length(x)
for j = 1:length(y)
input_vector(1) = x(i,j);
input_vector(2) = y(i,j);
f(i,j) = stalagmite(input_vector); %Values of function in given search space
end
end
surfc(x,y,-f); %Generating the surface
shading interp % To remove the lines produced by the meshgrid
% Statistical behaviour (Unbounded)
tic % Calculating the study time using the command tic and toc
for i = 1:num_cases
[inputs,fopt(i)] = ga(@stalagmite,2); %Calculating the optimized value of the function
xopt(i) = inputs(1); % at each iteration
yopt(i) = inputs(2);
end
study1_time = toc %Displaying the time taken to evaluate
% Plotting the graph
figure(1)
subplot(2,1,1)
surfc (x,y,-f);
hold on
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
xlabel('x');
ylabel('y');
grid on
subplot(2,1,2)
plot(-fopt);
xlabel('iterations')
ylabel('function min value')
% Statistical Behaviour (Bounded)
tic % Calculating the study time through tic and toc command
for j = 1:num_cases
[inputs,fopt(j)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]); % Calculating the optimized value of the function
xopt(j)=inputs(1); % At each iteration with Boundation
yopt(j)=inputs(2);
end
study2_time=toc
% Plotting
figure(2)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
xlabel('x');
ylabel('y');
grid on
subplot(2,1,2)
plot(-fopt);
xlabel('iterations')
ylabel('function min value')
options = optimoptions('ga','populationsize',300) %Increasing the size of population by using the optimoptions function
tic
for k = 1:num_cases
[inputs,fopt(k)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);
xopt(k)=inputs(1);
yopt(k)=inputs(2);
end
study3_time = toc
% Plotting
figure(3)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r');
xlabel('x');
ylabel('y');
grid on
subplot(2,1,2)
plot(-fopt);
xlabel('iterations')
ylabel('function min value')
CONCLUSION
ga is useful for optimizing the function but still the result are not accurate as the maximum value for the functionn lies at -1,to overcome that using the given programme shown below.
NOTE: gaplotbestf,which plots the best and mean score of the population at every generation.Also setting gaplotstoppinng,which plots the percentage of stopping criteria satisfied.
clear all
close all
clc
%Creating the mesh
x = linspace (0,0.6,150);
y = linspace (0,0.6,150);
[xx,yy] = meshgrid(x,y);
% Plotting the surface of stalagmite
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
% Evaluating the function
options = optimoptions('ga','plotfcn',{@gaplotbestf,@gaplotstopping},'populationsize',170);
[x,fval] = ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],options)
xopt = x(1);
yopt = y(2);
figure(1)
hold on
plot3(xopt,yopt,-fval,'ro','markersize',5,'markerfacecolor','r')OUTPUT
It can be much more visualized that how the fitness is been calculated and best result is being displaced even with the decrease in size of population.