Week 4.1 - Genetic Algorithm
AIM:
To write the code to calculate the global maxima of the stalagmite function and to explain the syntax for genetic algorithm(ga)in Matlab.
OBJECTIVE:
- To write a code in MATLAB to optimise the stalagmite function and find global maxima of the function.
- To explain the concept of genetic algorithm and explain the syntax for ga in MATLAB.
- To plot the graph of all three studies and also for global maxima.
THEORY
STALAGMATE FUNCTION:
Stalagmite function is a mathematical model that helps to generate maximal and minimal.Global maxima is a maximum peak value in entire function and global minima is a minimum peak value in entire function.But local maxima and local minima are multiple number of values.
Optimisation is finding of maxima and minima is particular function but in genetic algorithm its only to find minima.
Here ,we have to optimise the stalagmite function using genetic algorithm,
GENETIC ALGORITHM
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.
Notion of Natural Selection
The process of natural selection starts with the selection of fittest individuals from a population. They produce offspring which inherit the characteristics of the parents and will be added to the next generation. If parents have better fitness, their offspring will be better than parents and have a better chance at surviving. This process keeps on iterating and at the end, a generation with the fittest individuals will be found.
This notion can be applied for a search problem. We consider a set of solutions for a problem and select the set of best ones out of them.
Five phases are considered in a genetic algorithm.
- Initial population
- Fitness function
- Selection
- Crossover
- Mutation
Initial Population
The process begins with a set of individuals which is called a Population. Each individual is a solution to the problem you want to solve.
An individual is characterized by a set of parameters (variables) known as Genes. Genes are joined into a string to form a Chromosome (solution).
In a genetic algorithm, the set of genes of an individual is represented using a string, in terms of an alphabet. Usually, binary values are used (string of 1s and 0s). We say that we encode the genes in a chromosome.
Fitness Function
The fitness function determines how fit an individual is (the ability of an individual to compete with other individuals). It gives a fitness score to each individual. The probability that an individual will be selected for reproduction is based on its fitness score.
Selection
The idea of selection phase is to select the fittest individuals and let them pass their genes to the next generation.
Two pairs of individuals (parents) are selected based on their fitness scores. Individuals with high fitness have more chance to be selected for reproduction.
Crossover
Crossover is the most significant phase in a genetic algorithm. For each pair of parents to be mated, a crossover point is chosen at random from within the genes.
For example, consider the crossover point to be 3 as shown below.
Offspring are created by exchanging the genes of parents among themselves until the crossover point is reached.
The new offspring are added to the population.
.
Mutation
In certain new offspring formed, some of their genes can be subjected to a mutation with a low random probability. This implies that some of the bits in the bit string can be flipped.
Mutation occurs to maintain diversity within the population and prevent premature convergence.
Syntax for GA
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options)
[x,fval,exitflag,output] = ga(___)
[x,fval,exitflag,output,population,scores] = ga(___)
Description
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.
[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.
MATHEMATICAL EQUATION :
PROGRAM :
For plotting global maxima
clear all
close all
clc
%defining our search space
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);
% creating a 2Dimensiuonal mesh
[xx yy] = meshgrid(x,y);
%Evaluating the stalagmite function
for i = 1:length(xx)
for j = 1:length(yy)
f(i,j) = stalagmite([xx(i,j),yy(i,j)]);
end
end
surfc(xx,yy,f)
shading interp
xlabel('x value')
ylabel('y value')
[x,fval] = ga(@stalagmite,2)
A function calling program of stalagmite function,
%stalagmite function
function[f] = stalagmite(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
A program to optimise the stalagmite function and finding of the global maxima of a function.
clear all
close all
clc
x = linspace (0,0.6,150);
y = linspace (0,0.6,150);
num_cases = 50
% creating a 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) = stalagmite(input_vector);
end
end
%study 1 - statistical behaviour
tic
for i = 1:num_cases
[inputs,fopt(i)] = ga(@stalagmite,2);
xopt(i) = inputs(1);
yopt(i) = inputs(2);
end
toc;
%plotting
figure(1)
subplot(2,1,1);
hold on
surfc(x,y,f );
shading interp;
xlabel('x data');
ylabel('y data');
plot3(xopt,yopt,fopt,'marker','o','MarkerSize',5,'MarkerFaceColor','y')
title('unbounded inputs')
subplot ( 2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('Function maximum')
%study 2 - bounding the values
tic
for i = 1:num_cases
[inputs,fopt(i)] = ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]);
xopt(i) = inputs(1);
yopt(i) = inputs(2);
end
toc;
%plotting
figure(1)
subplot(2,1,1);
hold on
surfc(x,y,f );
shading interp ;
xlabel('x data');
ylabel('y data');
plot3(xopt,yopt,fopt,'marker','o','MarkerSize',5,'MarkerFaceColor','y')
title('Bounded inputs')
subplot ( 2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('Function maximum')
%study 3 - Bounded inputs with increased population size
tic
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
toc;
figure(1)
subplot(2,1,1)
hold on
surfc(x,y,f );
shading interp;
xlabel('x data');
ylabel('y data');
plot3(xopt,yopt,fopt,'marker','o','MarkerSize',5,'MarkerFaceColor','y')
title('Bounded inputs with increased population size')
subplot ( 2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('Function maximum')
OUTPUT :
Plot for general stalagmite model :
Case 1 : Unbounded condition
Case 2 : Bounded condition
Case 3 : Bounded with increased population size
RESULT :
- The code for global maxima and stalagmite function is done.
- The concept of genetic algorithm and syntax is done.
- The graph for all three studies and global maxima are plotted.
ERRORS :
No errors occured while coding
REFERENCES :
https://in.mathworks.com/videos/what-is-a-genetic-algorithm-100904.html
https:i//in.mathworks.com/help/gads/ga.html