Week 4.1 - Genetic Algorithm

AIM: MATLAB CODE to optimise the stalagmite function and find the global maxima of the function.

THEORY:

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.

Optimization- Optimization refers to finding the values of inputs in such a way that we get the “best” output values. The definition of “best” varies from problem to problem, but in mathematical terms, it refers to maximizing or minimizing one or more objective functions, by varying the input parameters.

The set of all possible solutions or values which the inputs can take make up the search space. In this search space, lies a point or a set of points which gives the optimal solution. The aim of optimization is to find that point or set of points in the search space.

Stalagmite function-

1 .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.

2. 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'. It is a programming which does a selection of the best element with regard to some criterion from some set of population.

4. Optimization is done by systematically choosing input values ​​from within an allowed set and computing the results of the function.

5. In matlab to perform optimization we use 'Genetic Algorithm 'method which is a inbuild function in matlab to find the global maxima of that given stalagmite function.

Genetic Algorithms: Nature has always been a great source of inspiration to all mankind. Genetic Algorithms (GAs) are search based algorithms based on the concepts of natural selection and genetics. GAs are a subset of a much larger branch of computation known as Evolutionary Computation. GAs were developed by John Holland and his students and colleagues at the University of Michigan, most notably David E. Goldberg and has since been tried on various optimization problems with a high degree of success.

Process 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.

1. Initial population

2. Fitness function

3. Selection

4. Crossover

5. Mutation

1.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 Os). We say that we encode the genes in a chromosome.

2.Fitness fuction- Fitness Function (also known as the Evaluation Function) evaluates how close a given solution is to the optimum solution of the desired problem. It determines how fit a solution is.

• why we use fitness functions-

In genetic algorithms, each solution is generally represented as a string of binary numbers, known as a chromosome. We have to test these solutions and come up with the best set of solutions to solve a given problem. Each solution, therefore, needs to be awarded a fitness score, to indicate how close it came to meeting the overall specification of the desired solution. This score is generated by applying the fitness function to the test, or results obtained from the tested solution.

3.Selection-

The selectionj is based on the fitness score.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.

4.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. Offspring are created by exchanging the genes of parents among themselves until the crossover point is reached.

5. 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. The population has a fixed size. As new generations are formed, individuals with least fitness die, providing space for new offspring. The sequence of phases is repeated to produce individuals in each new generation which are better than the previous generation. Ga as function of matlab

Syntax for genetic algorithm-

we can give the inputs by any one of the following ways-

Description of input variables-

These are the various input syntaxes that can be given-

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.

Passing Extra Parameters (Optimization Toolbox) explains how to pass extra parameters to the objective function and nonlinear constraint functions, if necessary.

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,[],[],lb,ub,nonlcon,IntCon) or x = ga(fun,nvars,A,b,[],[],lb,ub,nonlcon,IntCon,options) requires that the variables listed in IntCon take integer values.

When there are integer constraints, ga does not accept linear or nonlinear equality constraints, only inequality constraints.

x = ga(problem) finds the minimum for problem, where problem is a structure.

Outputs from the functions-

[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

stalagmite function-

function A = stagalmite(input_vector)
x=input_vector(1);
y=input_vector(2);
f1=(sin(5.1*pi*x+0.5)).^6;
f2=exp((-4*log(2)*((x-0.0667).^2))/0.64);
f3=(sin(5.1*pi*y+0.5)).^6;
f4=exp((-4*log(2))*((y-0.0667).^2)/0.64);
A =-(f1*f2*f3*f4);
end

explanation- we store the outputs into results which contain maxima and fopt which contain the value at which the maxima is obtained.

main code-

close all
clear all
clc
%input
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
num_spaces=50;

%forming 2d array
[xx,yy]=meshgrid(x,y);
for i=1:length(x)
    for j=1:length(y)
        input_vector(1)=xx(i,j);
        input_vector(2)=yy(i,j);
        f(i,j)=stagalmite(input_vector);
    end
end
%study 1-unbounded
tic
for i=1:num_spaces
    [results,fopt(i)]=ga(@stagalmite,2)
    xopt(i)=results(1)
    yopt(i)=results(2)
end
study_time1=toc
figure(1)
subplot(2,1,1)
surf(x,y,-f)
shading interp
hold on
plot3(xopt,yopt,-fopt,'marker','o')
subplot(2,1,2)
plot(-fopt)
xlabel('iteration')
ylabel('function maximum')
%study 2- bounded
tic
for i=1:num_spaces
    [results,fopt(i)]=ga(@stagalmite,2,[],[],[],[],[0;0],[0.6;0.6])
    xopt(i)=results(1)
    yopt(i)=results(2)
end
study_time2=toc
figure(2)
subplot(2,1,1)
surf(x,y,-f)
shading interp
hold on
plot3(xopt,yopt,-fopt,'marker','o')
subplot(2,1,2)
plot(-fopt)
xlabel('iteration')
ylabel('function maximum')
%study3- increased initial population
tic
options=optimoptions('ga')
options=optimoptions(options,'populationsize',250)
for i=1:num_spaces
    [results,fopt(i)]=ga(@stagalmite,2,[],[],[],[],[0;0],[0.6;0.6],[],[],options)
    xopt(i)=results(1)
    yopt(i)=results(2)
end
study_time3=toc
figure(3)
subplot(2,1,1)
surf(x,y,-f)
shading interp
hold on
plot3(xopt,yopt,-fopt,'marker','o')
subplot(2,1,2)
plot(-fopt)
xlabel('iteration')
ylabel('function maximum')

explanation-

close all

clear all

clc

step-1

%input

x=linspace(0,0.6,150);

y=linspace(0,0.6,150);

num_spaces=50;

explanation- Initialising the value of x and y and taking values between the interval 0 to 0.6

step-2

%forming 2d array

[xx,yy]=meshgrid(x,y);

for i=1:length(x)

for j=1:length(y)

input_vector(1)=xx(i,j);

input_vector(2)=yy(i,j);

f(i,j)=stagalmite(input_vector);

end

end

explanation- for loop is used for taking the values of i and j and the function "stagalmite" is called.

step-3

%study 1-unbounded

tic

for i=1:num_spaces

[results,fopt(i)]=ga(@stagalmite,2)

xopt(i)=results(1)

yopt(i)=results(2)

end

study_time1=toc

figure(1)

subplot(2,1,1)

surf(x,y,-f)

shading interp

hold on

plot3(xopt,yopt,-fopt,'marker','o')

subplot(2,1,2)

plot(-fopt)

xlabel('iteration')

ylabel('function maximum')

explanation- Now we store the results into xopt,yopt to plot the surface of x,y,-f to get 3D plot.

OUTPUT-

step-4

%study 2- bounded

tic

for i=1:num_spaces

[results,fopt(i)]=ga(@stagalmite,2,[],[],[],[],[0;0],[0.6;0.6])

xopt(i)=results(1)

yopt(i)=results(2)

end

study_time2=toc

figure(2)

subplot(2,1,1)

surf(x,y,-f)

shading interp

hold on

plot3(xopt,yopt,-fopt,'marker','o')

subplot(2,1,2)

plot(-fopt)

xlabel('iteration')

ylabel('function maximum')

explanation- we store the another results for figure 2 to plot the surface x,y,-f to get 3d plot.

output-

step-5

%study3- increased initial population

tic

options=optimoptions('ga')

options=optimoptions(options,'populationsize',250)

for i=1:num_spaces

[results,fopt(i)]=ga(@stagalmite,2,[],[],[],[],[0;0],[0.6;0.6],[],[],options)

xopt(i)=results(1)

yopt(i)=results(2)

end

study_time3=toc

figure(3)

subplot(2,1,1)

surf(x,y,-f)

shading interp

hold on

plot3(xopt,yopt,-fopt,'marker','o')

subplot(2,1,2)

plot(-fopt)

xlabel('iteration')

ylabel('function maximum')

explanation- Again we store the new results into xopt,yopt to plot the surface of x,y,-f to get the 3d plot.

output-

CONCLUSION- The global maxima of the function is found and plotted in 3D plot and the three output and explanation is given above.