Challenge

Week 4.1 - Genetic Algorithm

AAniket Jagtap5 Apr 2022

MATLAB

AIM-

To write a code in MATLAB to optimise the stalagmite function and find the global maxima of the function.
To clearly explain the concept of genetic algorithm and also explain the syntax for ga in MATLAB.
Plotting graphs for all 3 studies and for F maximum vs no. of iterations.

THEORY-

The genetic algorithm is a method for solving both constrained and unconstrained optimization problems that is based on natural selection, the process that drives biological evolution. The genetic algorithm repeatedly modifies a population of individual solutions.
At each step, the genetic algorithm selects individuals from the current population to be parents and uses them to produce the children for the next generation.
Over successive generations, the population "evolves" toward an optimal solution.
ou can apply the genetic algorithm to solve a variety of optimization problems that are not well suited for standard optimization algorithms, including problems in which the objective function is discontinuous, nondifferentiable, stochastic, or highly nonlinear.
The genetic algorithm can address problems of mixed integer programming, where some components are restricted to be integer-valued.
Flow chart below outlines the main algorithmic steps -

In this study we are considering 3 cases -
1. Study of GA using random sampling.
2. Study of GA using lower and upper bound.
3. Study of GA using lower and upper bound and increasing population size

PROGRAM/CODE-

Below is the code to optimise the stalagmite function and find the global maxima of the function.

clear all
close all
clc

x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
num_cases = 50


for i = 1:length(x)
    for j=1:length(y)
        input_vector(1) = x(i);
        input_vector(2) = y(j);
        f(i,j)=stalagmite(input_vector);
    end
end

%Study 1 -Statistical Behaviour
tic
for k = 1:num_cases
    [inputs, fopt(k)] = ga(@stalagmite,2)
    xopt(k)=inputs(1)
    yopt(k)=inputs(2)
    
end

study1_time = toc

figure (1)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('Unbounded Inputs')
subplot(2,1,2)
plot(-fopt)
xlabel('Iterations')
ylabel('Function Maximum ')

%Study 2 - Statistical behavior with upper and lower bounds
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

study2_time = toc
figure (2)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('Bounded Inputs')
subplot(2,1,2)
plot(-fopt)
xlabel('Iterations')
ylabel('Function Maximum ')

%Study 3 - Increasing GA iterations
options = optimoptions('ga');
options = optimoptions(options,'PopulationSize',170);

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
   
study3_time = toc
figure (3)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('Bounded Inputs with modified polulation size')
subplot(2,1,2)
plot(-fopt)
xlabel('Iterations')
ylabel('Function Maximum ')

Code 1

function [f] = stalagmite(input_vector)
f1x=(sin(5.1*pi*x+0.5))^6
f1y=(sin(5.1*pi*y+0.5))^6

f2x= exp(-4*log(2)*(x-0.667)^2)/0.64
f2y= exp(-4*log(2)*(y-0.667)^2)/0.64

f = -(f1x*f1y*f2x*f2y)

end

Code 2 - Stalagmite Function code

CODE EXPLAINATION -

Firstly, with the help of linspace command we are defining our search space area and then finalizing the number of cases or iterations to be 50 for this study.
Using for loop, we are creating an array of x and y values and then defining stalagmite function for plotting the stalagmite mathematical model.
The mathematical equation which we have used for defining our function is as below-
In the first study which is considering random sampling, we are using GA syntax as - [x,fval] = ga(@fun,nvars)
- where x = inputs, fval = Minimum function value, fun = Function and nvars = Number of variables.
- Note - GA function always gives minimum value as the output
- By running the array of values in the for loop, we will get values of x and y where the function value is optimum.
- By plotting those values, using subplot and plot3 command we are getting our mathematical model and function values
- For results, see Figure (1)
- Since, GA gives minimum value and our aim is to find maximum function values, we just put (-ve) sign ahead of fopt and the values are plotted in reverse direction.
In the second study which is considering the lower and upper bound sampling, we are using GA syntax as - [x,fval]=ga(@fun,nvars,A,B,Aeq,Beq,lb,ub)
- where lb = lower bound and ub = upper bound
- In this study, since we have defined our lower and upper bounds the GA will have limited interpolations to make and find the maximum function value.
- For results, see Figure (2)
In the third study which is considering the lower and upper bound sampling with increased population size we are using GA syntax as - [x,fval]=ga(@fun,nvars,A,B,Aeq,Beq,lb,ub,nonlcon,IntCon,options)
- In this study since we have increased the population size, the GA gets more options and we get a more preciser function value.
- For results, see Figure (3)
For all the cases we have used shading interp command so that we can get a smoother 3D curve (Stalagmite model)
We have also used Tic and Toc command to calculate the study time. Study 3 took maximum time out of all studies.