Week 4.1 - Genetic Algorithm
AIM:
1. Understand and explain the concept of genetic algorithm and brief about different syntaxes in ga.
2. Plot graphs for all 3 studies and for F maximum vs no. of iteration.
THEORY:
A genetic algorithm (GA) is a metaheuristic(Solution approach) inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). 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.
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 Algoritham-
- 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 a new population based on no. of iterations performed.
1. Fitness Function
It is that function that we have to optimize and its optimization is done with the help of a number of generations that have been formed by the code in accordance with the fitness value of each member generated, these fitness values are used for the reproduction of the children.
2. 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 an unfocused search.
2. Reproduction
Generation of offspring happen in 2 way:
A.)Crossover
B.)Mutation
2. A) 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.
2. B) 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.
The syntax for GA:
x = ga(fun,nvars), finds a local unconstrained minimum, x, to the objective function, fun. nvars is the dimension (number of design variables) of fun.
x = ga(fun,nvars,A,b) finds a local minimum x to fun, subject to the linear inequalities A*x ≤ b. ga evaluates the matrix product A*x as if x is transposed (A*x').
x = ga(fun,nvars,A,b,Aeq,beq) finds a local minimum x to fun, subject to the linear equalities Aeq*x = beq and A*x ≤ b. (Set A=[] and b=[] if no linear inequalities exist.) ga evaluates the matrix product Aeq*x as if x is transposed (Aeq*x').
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that a solution is found in the range lb ≤ x ≤ ub. (Set Aeq=[] and beq=[] if no linear equalities exist.)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon) subjects the minimization to the constraints defined in nonlcon. The function nonlcon accepts x and returns vectors C and Ceq, representing the nonlinear inequalities and equalities respectively. ga minimizes the fun such that C(x) ≤ 0 and Ceq(x) = 0. (Set lb=[] and ub=[] if no bounds exist.)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options) minimizes with the default optimization parameters replaced by values in options. (Set nonlcon=[] if no nonlinear constraints exist.) Create options using optimoptions.
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon) or x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options) requires that the variables listed in intcon take integer values.
x = ga(problem) finds the minimum for problem, a structure described in problem.
[x,fval] = ga(___), for any previous input arguments, also returns fval, the value of the fitness function at x.
[x,fval,exitflag,output] = ga(___) also returns exitflag, an integer identifying the reason the algorithm terminated, and output, a structure that contains output from each generation and other information about the performance of the algorithm.
example
[x,fval,exitflag,output,population,scores] = ga(___) also returns a matrix population, whose rows are the final population, and a vector scores, the scores of the final population.
BODY:
We first write a function code to define the stalagmite function
function [f] = stalagmite(input_vector) %name the function
x = input_vector(1); % data set for different values of x
y = input_vector(2); % data set for different values of
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);
endWhile using the function at the end as we are using the -ve sign, the plot will be reversed as we have to find the maxima of the given function using ga.
INSTRUCTIONS:
1. Firstly, create the mesh from the two vectors which will be acting as a search space to find the maxima of the function.
2. 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.
3. Now calculate the values of fitness function at each point of the mesh grid formed.
4. Now create its surface through ‘surfc’ command which will also provide the contour of the generated surface.
5. While using surfc applying the -ve 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 a positive value for the maxima using the -ve sign.
A.)STATISTICAL STUDY 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 (which gives the inputs ie. Values of x and y at which the stalagmite function is minimum)
Then plotting it into the surf command and also providing the optimum values of the function that how it is varying.
Also, use the tic and toc command to evaluate the time used for optimizing the function.
From the figure, as shown above, it is clear that the results are not that accurate and give a lot of variation in optimizing the function.
B.)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.
It is clearly seen from the above that after providing the boundary limits the variation in the optimum value of the function has reduced.
c.)STATISTICAL STUDY WITH INCREASE IN SIZE OF POPULATION
This study using the ‘optimoptions’ command to increase the size of the population.
The benefit of that will be it will be working on a large set of numbers, which will increase the possibilities of finding the better fitness of the function.
The drawback is that the time for evaluation will increase.
Total code:
clear
close all
clc
%% PROVIDING THE SEARCH SPACE
x=linspace(0,0.8,200);
y=linspace(0,0.8,200);
[X,Y]=meshgrid(x,y);
num_cases = 150; %Number of iterations for which the function will evaluate the fitness
%% SURFACE PLOT OF STALAGMITE FUNCTION
for i=1:length(X) % starting from variable i through all values of x
for j=1:length(Y) % starting from variable j through all values of 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 mesh grid
%% STATISTICAL BEHAVIOUR (UNBOUNDED)
tic %calculating the study time through tic and toc command
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); % to create countour of surface
hold on
shading interp % interp is to interpolate and obtain values
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')
%% MODIFYING THE NUMBER OF POPULATION
options=optimoptions('ga','populationsize',300) % increaing the size of population by using
% 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')KEYNOTE: One main difference to be observed between bounded and unbounded studies is in the former case, the value strictly lies in the specified range (0,0.8), which does not happen in the latter case.
ERROR:
use of proper syntax for ga , without @stalagmite, the error is shown as follows:
References: