Week 4.1 - Genetic Algorithm

• 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(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options)

• 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.

MATLAB CODES -

1. Stalagmite Function code -

function [S] = stalagmite_max(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*0.6931)*((x-0.0667).^2/0.64));
f2_y = exp((-4*0.6931)*((y-0.0667).^2/0.64));

S = -(f1_x * f1_y * f2_x * f2_y);
end

Explanation -

• In this function code we calculate the values of x,y to get the stalagmite value by which we cam get the global maxima.

2. Final code to optimise the stalagmite function and find the global maxima of the function -

clear all
close all
clc

%defining research space
x= linspace(0,0.6,150);
y= linspace(0,0.6,150);

num_cases = 100;

%creating a 2D mesh
[xx yy] =meshgrid(x,y);

%evaluating 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_max(input_vector);
    end
end

%study1 stastical behavior
tic
for i=1:num_cases
    [inputs,fopt(i)] = ga(@stalagmite_max,2);
    xopt(i) = inputs(1);
    yopt(i) = inputs(2);
end

study1_time = toc

fopt=-fopt

figure(1)
subplot(2,1,1);
hold on
surfc(xx,yy,-f)
shading interp
plot3(xopt,yopt,fopt,'Marker','o','markersize',5,'MarkerFaceColor','r')
title('Unbounded Inputs')
subplot(2,1,2)
plot(fopt)
xlabel('Iteration')
ylabel('Function Maximum')

%study2 stastical behavior with upper and lower bounds
tic
for i=1:num_cases
    [inputs,fopt(i)] = ga(@stalagmite_max,2,[],[],[],[],[0;0],[1;1]);
    xopt(i) = inputs(1);
    yopt(i) = inputs(2);
end

study2_time = toc

fopt=-fopt

figure(2)
subplot(2,1,1);
hold on
surfc(xx,yy,-f)
shading interp
plot3(xopt,yopt,fopt,'Marker','o','markersize',5,'MarkerFaceColor','r')
title('Bounded Inputs')
subplot(2,1,2)
plot(fopt)
xlabel('Iteration')
ylabel('Function Maximum')

%study3 stastical behavior with more GA iterations
options = optimoptions('ga');
options = optimoptions(options,'PopulationSize',500);

tic
for i=1:num_cases
    [inputs,fopt(i)] = ga(@stalagmite_max,2,[],[],[],[],[0;0],[1;1],[],[],options);
    xopt(i) = inputs(1);
    yopt(i) = inputs(2);
end

study3_time = toc

fopt=-fopt

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')
subplot(2,1,2)
plot(fopt)
xlabel('Iteration')
ylabel('Function Maximum')

Explanation & Results -

• First we initialise the values of x and y taking 150 values between the interval 0 and 0.6, We consider no of iterations 100.

• Since we need to plot a 3D surface plot we take the 2d array of x and y using meshgrid command and store them in input_vector array ,so that it can be used as an argument to call the function stalagmite we create another function file for the stalagmite function. here we take the ouput as -f since ga only minimises a given function.so if -f is considered it automatically gives us the maxima.

• Now for Study case 1 -

We take num_cases value and we use the genetic algorithm function to caluclate the maxima of the function. we store the outputs into results which contain maximas and fopt which contain the value at which the maxima is obtained.

Here the ga is initiated with random population.

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

We use the plot3 command to make a plot of xopt,yopt and -fopt to plot the optimum maximas.

Also let us plot the fopt function so that we can visualise the optimum values at each iteration.

To get the time taken to complete cycle we used the tic toc commands to record the time.

The executed study 1 output is displayed as -

Here we can see the optimal values went outside the values of x and y resulting in falling away from stagalmite surface.

So we failed to reach the global maxima in this case.

the study 1 time = 160.3841s

• Now for Study case 2 -

let us specify the lower and upper bounds of x and y in this case,the ga is initiated with random population only.

Then we ploted the same way as study case 1.

The executed study 2 output is displayed as -

Though the optimum values are within the limits it could not give us the exact maxima values.

The study 2 time = 141.8880s

• Now for Study case 3 -

In this case let us increase our initial population which is default as 100 to 500.

Here in this case to set a initial population we need to use command called optimumoptions and get our value of 'populationsize' to 500.

Then we ploted the same way as previous plots.

The executed study 3 output is displayed as -

Though the time of execution is more in this case we are sucessfully to find out the correct global maxima for the stalagmite function.

So by increasing the initrial population size of the ga we can increase the accuracy of the algorithm.

The study 3 time = 1.2984e+03 s which is more than above two studies.