Week 4.1 - Genetic Algorithm
Aim write a code in matlab to optimise the stalagmite function and find the global maxima of the function
Objective
- Clearly explain the concept of genetic algorithm
- Explain syntax for ga
- Plot graphs for all 3 studies for F Maximum vs. no of iterations
Introduction
Optimization Optimization is the act of obtaining the best result under a given circumstances. Optimization is the mathematical discipline which is concerned with finding the maxima and minima of functions
Stalagmite function 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. Stalagmite showing function maximum and minimum (Global & Local)
Genetic algorithm 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
Termination
The algorithm terminates if the population has converged (does not produce offspring which are significantly different from the previous generation). Then it is said that the genetic algorithm has provided a set of solutions to our problem.
difference between Genetic Algorithm and Traditional Algorithm
|
Genetic Algorithm |
Traditional Algorithm |
|
Genetics and natural selection to solve optimization problem |
Step by step procedure to solve a given problem |
|
More Advanced |
Not as advanced |
|
Used in field such as Machine learning and artificial intelligent |
Used to fields such as programming, mathamatics |
|
Probabilistic Rules |
Fully deterministic rules |
|
Search solution on a population of point |
Search solution on single point |
Syntax for Genetic Algorithm
- x = ga(fun,nvars)
- x = ga(fun,nvars,A,b)
- x = ga(fun,nvars,A,b,Aeq,beq)
- 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 = ga(problem)
- [x,fval] = ga(___)
- [x,fval,exitflag,output] = ga(___)
- [x,fval,exitflag,output,population,scores] = ga(___)
- ga(fun,nvars) finds a local unconstrained minimum, x, to the objective function, fun. nvars is the dx = 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.imension (number of design variables) of fun.
- 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.
Program (code) to solve genetic algoritham
- code for Stalagmite function
function [f]=func(input_vector)
a1=(sin(5.1*pi*input_vector(1)+0.5))^6;
a2=(sin(5.1*pi*input_vector(2)+0.5))^6;
a3=exp(-4*log(2)*((input_vector(1)-0.0667)^2)/(0.64));
a4=exp(-4*log(2)*((input_vector(2)-0.0667)^2)/(0.64));
f=-(a1*a2*a3*a4);
end
- code for Genetic algoritham
close all
clear all
clc
%input
x=linspace(0,0.6,150)
y=linspace(0,0.6,150)
num_cases=60
[xx yy]= meshgrid(x,y);
%fuction calling
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)=func(input_vector);
end
end
%case1 unbounded inputs
tic
for i=1:num_cases
[inputs,fopt(i)]=ga(@func,2);
xopt(i)=inputs(1);
yopt(i)=inputs(2);
end
time1=toc
% case1
figure(1)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
xlabel('iterations in x ')
ylabel('function maxima')
zlabel('iterations in z')
plot3(xopt,yopt,-fopt,'marker','o','MarkerSize',5,'MarkerFaceColor','r')
legend('minima point','maxima point')
title('unbounded inputs')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('function maxima')
%case study 2 bounded inputs
tic
for i=1:num_cases
[inputs,fopt(i)]=ga(@func,2,[],[],[],[],[0,0],[0.5,0.5]);
xopt(i)=inputs(1);
yopt(i)=inputs(2);
end
time2=toc
%bounded inputs
figure(2)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
xlabel('iterations in x ')
ylabel('function maxima')
zlabel('iterations in z')
plot3(xopt,yopt,-fopt,'marker','o','MarkerSize',5,'MarkerFaceColor','r')
legend('minima point','maxima point')
title('bounded inputs')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('function maxima')
%case study 3 optimisation
options=optimoptions('ga');
options=optimoptions(options,'PopulationSize',200);
tic
for i=1:num_cases
[inputs,fopt(i)]=ga(@func,2,[],[],[],[],[0;0],[0.5;0.5],[],[],options);
xopt(i)=inputs(1);
yopt(i)=inputs(2);
end
time3=toc
%optimisation
figure(3)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
xlabel('iterations in x ')
ylabel('function maxima')
zlabel('iterations in z')
plot3(xopt,yopt,-fopt,'marker','o','MarkerSize',5,'MarkerFaceColor','r')
legend('minima point','maxima point')
title('optimisation')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('function maxima')
Step wise explanation of code
1.Stalagmite function
as we discusson the theory of the stalagmite function to solved the equvation we create new strip in matlab and creating the function ''func'' store in array [f] then sliting the the Stalagmite function in to a1,a2,a3,a4 and assign to the f puting -ve because ga calculating mimima only
2.Genetic algoritham
- 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.) using this ga syntax we gate 4 maxima ponit then again using following ga syntax to find one maxima point
- 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. using this ga syntax we get the final maxima point
RESULT
