Genetic Algorithm: Genetic Algorithm is based on evolution theory proposed by Charles Darwin. It is a natural selection process Genetic algorithm always returns the minimum values. we can use Genetic Algorithm for problems where there is a clear way to evaluate fitness. If your search space is not well constrained or your evaluation process is computationally expensive, discontinuous objective function, nondifferentiable, stochastic, or highly nonlinear. Genetic Algorithms may not find solutions in a sane amount of time. GAs rely on biologically inspired processes crossover, mutation and, selection.
Process followed by Genetic Algorithm:
Genetic Alogorithm starts with population initialization, the sample data in the population undergoes three steps
1.Selection: In selection process fittest individuals i.e the parents are selected from the sample data. The more fitter individuals in the population produce fitter offsprings over a time.
2.Crossover: crossover is used to combine the genetic information of two parents to generate new offspring. for example Set of genes from parent A is combined with set of genes with parent B. A better offsprings with both the parents qualities are produced.
3.Mutation: mutation is used to maintain genetic diversity from one generation of a population of genetic algorithm chromosomes to the next. If mutation is not performed it results in convergence and local minima. It is analogous to biological mutation. Mutation alters one or more gene values in a chromosome from its initial state.
STALAGMITE Function: Stalagmite and stalactite are mathematical functions that resembles Stalagmite and Stalactite structures in nature. Stalagmite is formed on the cave floor and Stalactite is formed from the roof of the cave. These formations are formed due to redeposition of minerals from water droplets which evaporates leaving behind minerals. When stalactite and stalagmite meet it is called Stalagnates.
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,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,[],[],lb,ub,nonlcon,Intcon,options) minimizes with the default optimization parameters replaced by values in options. (Set nonlcon=[] if no nonlinear constraints exist.) Create options using optimoptions.
Code Explaination:
1.Defining the search space for the GA to be performed. Using the mesh caommand to create 2d matrix.Writing stalagmite function which will be called in the main function.
2.Negative sign is used in the stalagmite function to get the global maximum as GA always returns global minima.
3.Assigning the values to inputs using for loop and to be used in the stalagmite function .
4.Getting values of stalagmite to the variable f.
5.Applying Genetic algorithm on stalagmite function.
6.Assigning the Genetic algoritm values to inputs, and then assigning to xopt,yopt ,foptusing for loop.
7.Three studies are made to get the optimised results.in the first study bounds are not defined. In the second study boundaries are defined. In the third study population size is defined.
8.Used tic and toc commands to know the time taken for each case.
9.creating surface using surf command among xx,yy,f ,using plot3 command to plot xopt,yopy,fopt. Plot fopt to show the optimized point.
Stopping criteria:
1. The iterations stops when the iteration limit is reached.
2. When it is known that there wont be any change in the result for the iteration .
3. When the population limit is reached.
%writing function for given stalagmate function
function f = stalagmite_func(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;
f2_x=exp(-4*log(2).*(x-0.0667)^2)/0.64;
f2_y=exp(-4*log(2).*(y-0.0667)^2)/0.64;
f =-(f1_x*f1_y*f2_x*f2_y);
end
% MAIN SCRIPT
close
clear
clc
% defining the search space
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);
% to create 2d mesh
[xx, yy] = meshgrid(x,y);
% % assigning valus to inputs
for i = 1:length(xx)% takes values until length of xx
for j = 1:length(yy)% takes values until length of xx
inputs(1) = xx(i,j);
inputs(2) = yy(i,j);
f(i,j) = stalagmite_func(inputs);
end
end
num_case = 50; % creating a variable and assigning the number to which iterations performed
tic % Starts the timer
% Studt case-1
for z = 1:num_case% takes values until length of num_cases
[inputs,fopt(z)] = ga(@stalagmite_func,2);% solve stalagmite function using GA and assigning to variables
xopt(z) = inputs(1);
yopt(z) = inputs(2);
end
Time_study1 = toc; % Gives time taken to perform the case
figure(1)
subplot(2,1,1) % creates sub-plot in the figure
hold on
surfc(xx,yy,-f) % Creates a surface
shading interp % varies the color in each line segment and face by interpolating the colormap index or true color value across the line or face.
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')% plot3 command used to plot in 3 axis
% marker is used to create a visible points on the plot
title('unbounded inputs')
xlabel('x-axis')
ylabel('y-axis')
zlabel('z-axis')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('Function maximum')
% Study Case-2 Defining upper and lower bounds
tic
for z = 1:num_case
[inputs,fopt(z)] = ga(@stalagmite_func,2,[],[],[],[],[0;0],[0.4;0.4]);
%defines a set of lower and upper bounds on the design variables
xopt(z) = inputs(1);
yopt(z) = inputs(2);
end
Time_study2 = toc;
figure(2)
subplot(2,1,1)
hold on
surfc(xx,yy,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','b')
title('Bounded inputs')
xlabel('x-axis')
ylabel('y-axis')
zlabel('z-axis')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('Function maximum')
clc
% Study Case-3 Increasing iterations
options = optimoptions('ga');% returns a set of default options for the SolverName solver.
options = optimoptions(options,'PopulationSize',270);%Defining population size
tic
for z = 1:num_case
[inputs,fopt(z)] = ga(@stalagmite_func, 2,[],[],[],[],[0;0],[0.4;0.4],[],[],options);
% takes the values of options and solves ga in required population
xopt(z) = inputs(1);
yopt(z) = inputs(2);
end
Time_study3 = toc;
figure(3)
subplot(2,1,1)
hold on
surfc(xx,yy,-f)
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('Increased population')
xlabel('x-axis')
ylabel('y-axis')
zlabel('z-axis')
subplot(2,1,2)
plot(-fopt)
xlabel('iterations')
ylabel('Function maximum')
% The Global Maximum at the position
display(inputs,'Global maximum')