Week 5 - 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,[],[],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.

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 CODE:

Stalagmite function:

function [f] = stalagmite_f(input_vector)

x = input_vector(1);
y = input_vector(2);

f_1_x = (sin(5.1*pi*x + 0.5))^6;
f_1_y = (sin(5.1*pi*y + 0.5))^6;
f_2_x = exp(-4*log(2)*((x-0.0667)^2/0.64));
f_2_y = exp(-4*log(2)*((y-0.0667)^2/0.64));
[f] = -((f_1_x)*(f_2_x)*(f_1_y)*(f_2_y));  % -ve sign is used to obtain the peak value.

end

Main code:

clear all
close all
clc

% Inputs
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);

num_cases = 50; % to run ga 50 times

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

% Evaluating the 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_f(input_vector);
    end
end

% Study 1 - Statistical Behavior
tic
for i=1:num_cases
    [inputs, fopt(i)] = ga(@stalagmite_f,2);
    xopt(i)=inputs(1);
    yopt(i)=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',3,'markerfacecolor','r')
title('Unbounded Inputs')

subplot(2,1,2)
plot(fopt)
xlabel('Number of iterations')
ylabel('Function Maximum')

% Study 2 - Statistical Behaviour  with Upper & Lower Bounds
for i = 1:num_cases
    [inputs, fopti] = ga(@stalagmite_f,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',3,'markerfacecolor','r')
title('Bounded Inputs')

subplot(2,1,2)
plot(fopt)
xlabel('Number of iterations')
ylabel('Function Maximum')

% Study 3 - Bounded Inputs with Increased Population

options=optimoptions('ga','populationsize',620);
tic
for i = 1:num_cases
    [inputs, fopt(i)] = ga(@stalagmite_f,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',3,'markerfacecolor','r')
title('Bounded Inputs with Increased Population Size')

subplot(2,1,2)
plot(fopt)
xlabel('Number of iterations')
ylabel('Function maximum')

Code Explanation:

• A Stalagmite function file is created, with the f value being -ve to determine the global maxima instead of the default global minima.

• Inputs 'x' and 'y' have 150 values from 0 to 0.6.

• Created a 2D array using meshgrid function.

• Ran ga 50 times.

• Using for loop, evaluated the stalagmite function.

• Study 1 – Statistical Behavior

  tic toc command is used to determine the total execution time.

  Initiated a for loop.

  Plotted figure 1, which contains two subplots for unbounded inputs.

• Study 2 – Statistical Behavior with Upper and Lower Bounds and Study 3 – with Increased Population were also conducted.

  Similarly, tic toc commands were used to determine the total execution time and figures, along with sub plots were plotted.

No major error occured during coding.

Results:

Study 1 plots:

Study 2 plots:

Study 3 plots:

Workspace:

Study time:

Conclusion:

• The concept of genetic algorithm (ga) has been explained.
• The syntax for ga in MATLAB has been explained.
• Successfully executed a code in MATLAB to optimize the stalagmite function and find the global maxima of the function.
• Successfully plotted graphs for all 3 studies.

References:

• au.mathworks.com. (n.d.). Find minimum of function using genetic algorithm - MATLAB ga - MathWorks Australia. [online] Available at: https://au.mathworks.com/help/gads/ga.html#mw_46d5ecc6-7ae1-4400-821b-3a29c14be603 [Accessed 3 Feb. 2023].
• au.mathworks.com. (n.d.). How the Genetic Algorithm Works - MATLAB & Simulink - MathWorks Australia. [online] Available at: https://au.mathworks.com/help/gads/how-the-genetic-algorithm-works.html [Accessed 3 Feb. 2023].
• GeeksforGeeks. (2017). Genetic Algorithms. [online] Available at: https://www.geeksforgeeks.org/genetic-algorithms/.