Week 4.1 - Genetic Algorithm

x = ga(fun,nvars)
x = ga(fun,nvars,A,b)
x = ga(fun,nvars,A,b,Aeq,beq)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon)
x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options)
x = ga(problem)
[x,fval] = ga(___)
[x,fval,exitflag,output] = ga(___)
[x,fval,exitflag,output,population,scores] = 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) 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(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.

where ga is an inbuild function by mathlab then providing the objective function then the dimensions.

in case we do not have the values of A,b,Aeq,beq,lb,ub,nonlcon,intcon,options we need to put [] in that space.

code to optimise the stalagmite function

clear all
close all
clc

%defining our search space
 x=linspace(0,0.6,150);
 y=linspace(0,0.6,150);
 num_cases=50;                                  % running ga 50 times

 %converting 2d plot of x and y which were in 1d
 [xx,yy]=meshgrid(x,y);

  for k=1:length(xx);                           % values of xx
     for j=1:length(yy);                        % values af yy
         input_vector(1)=xx(k,j);               %creating array  each values from xx 
         input_vector(2)=yy(k,j);               %creating array  each values from yy
         f(k,j)=stalagmite(input_vector);       % function creates for stalagmite in figure    
     end
 end

 figure(1)
surfc(xx,yy,-f)                                 % making graph figure from xx,yy,f values
shading("interp")                               % removing lines from the plot giving smooth look
xlabel("x value")                              
ylabel("y value")
zlabel("f values")
colorbar
title('stalagmite')

%study 1 satistical behaviour 
  tic                                            % time start 
 for i=1:num_cases;
     [inputs,fopt(i)]=ga(@stalagmite,2);
     xopt(i)=inputs(1);
     yopt(i)=inputs(2);
 end
study_1_time=toc                                      % end time stored in study_1

 figure(2)
 subplot(2,1,1)                                       % to attach two graph in one screen.
 hold on
surfc(x,y,f)
shading interp
% plotting the optimum points
plot3(xopt,yopt,fopt,Marker='o',MarkerSize=5,MarkerFaceColor='r')
title('unbounded inputs')
xlabel("x value")
ylabel("y value")
subplot(2,1,2)
plot(fopt)
xlabel("iteration")
ylabel("function minimum")
tic
% study 2-statistical behavior -with upper and lower bonds
 for i=1:num_cases;
     [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[0.6;0.6]);  % giving upper and lower boundry  for ga 
     xopt(i)=inputs(1);
     yopt(i)=inputs(2);
 end
study_2_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')
xlabel("x value")
ylabel("y value")
subplot(2,1,2)
plot(fopt)
xlabel("iteration")
ylabel("function minimum")

% study 3 increasing ga iterations
options =optimoptions("ga");                       
options=optimoptions(options,PopulationSize=400);           % changing initial population size so that we can get global minima
% instade of global local minima
tic
for i=1:num_cases;
     [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[0.6;0.6],[],[],options);
     xopt(i)=inputs(1);
     yopt(i)=inputs(2);
 end
study_3_time=toc


 figure(4)
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')
xlabel("x value")
ylabel("y value")
subplot(2,1,2)
plot(fopt)
xlabel("iteration")
ylabel("function minimum")

% creatin table for time
time_taken=[study_1_time;study_2_time;study_3_time]                      
time_taken_by=['study_1_time';'study_2_time';'study_3_time']
result=table(time_taken,time_taken_by)

function codes which is created

function f=stalagmite(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);
f=-(f1.*f2.*f3.*f4);
end

output plots

study 1 plot

study 2 plot

study 3 plot

time take by each graph

explanation of the code

first providing x values and y values using linspace command which are 1 diamention values.

then converting x and y values into xx and yy values using meshgrid command which are 2 dimention values

then using numcase this is to run ga 50 time.

creating the loop which will provide a stalagmite design in the graph

first loop take the values of xx and second loop take the values of yy and will be provided to input vectors

stalagmite is the function which is been created

then using surfc command to plot xx,yy -f values {-f is used to make the stalagmite figure not to be in inverse direction}

shading("interp") comand is used to make the plot look smooth an remove grid from it.

providing x label, ylabel,z label,title and colorbar.

now ,

study 1

where we are using ga command and providing stalagmit function then no in inputes vlues which are xopt and yopt

and running it for 50 time using of tic and toc command to not how much time does itake.

using subplot to plot two plot in figure

first plot is plot3 comand which shows stalagmit figure with random points

second plot is fopt which shows the values of -1 to 1 because boundries were not defined . and so study 1 gives n number of values.

study 2

running for 50 time using of tic and toc command to not how much time does itake.

in ga providing [] which are blank space , then providing upper and lower bondries we provided this because ga function was picking random valuse which was not defined and now you can see study 2 graph with very less points which are local maxima.

study 3

running for 50 time using of tic and toc command to not how much time does itake.

giving options=optimoptions(options,PopulationSize=400) means changing initial population size so that we can get global minima instade of global local minima as shown in study 2.

now every time well run the code will get the same values and

study 3 figure shows the value of -0.99999 whic opposit of the global maxima thats means the global maxima will be 0.9999

explanation of function

x and y are the input vector and f1,f2,f3,f4 are the formulas

f=-(f1.*f2.*f3.*f4) puting (-) because ga always provide the global minima and we need to find he globlal maxima to we use the (-) sign so the the answer wll be reverse and we can get global maxima instade of global minima

errors got in the program

this error was due to not putting - sign in function and it took 3 days me to understand

error 2

due to not putting @ before stalagmite because ga is a function and stalagmite in another function sowhen using function under another function need to add @.

error3

due to not providing this bracket [] because this brackes are used for a matrix opner

referance

https://in.mathworks.com/help/gads/ga.html

https://in.mathworks.com/discovery/genetic-algorithm.html

https://in.mathworks.com/videos/what-is-a-genetic-algorithm-100904.html

thank you