Week 4 - Genetic Algorithm

AIM:-

 To optimise the stalagmite function and find the global maxima of the function. 

 OBJECTIVE:-

 . expalin the concept of genetic algorithm in your own words and also explain the syntax for ga in MATLAB in your report.

  .Plot graphs for all 3 studies and for F maximum vs no. of iterations.

INTRODUCTION:-

. The genetic algorithm GA is a method for solving both constrained and unconstrained optimisation problem that is based on natural selection, the process that drives biologica evolution. GA repeatedly modified the populaton of individual solutions.

 .At each step,the genetic alogrithm selects individual at random from the current population to be parents and uses them to produce children for the next generation over successive generation, the population evolves towards an optimal solution.

 .GA can be applied to solved a variety of optimization problems that are not well suited for standard optimazation algorithms,including problems in which the objective function is discontinuous, non differentiable, stochastic, highly non linear.

 .The GA uses 3 main types of rules at each step to create the next generation from the current population:

            . Selection rule selects the individuals, called parents, that contribute to the population at the next generation.

            . crossover rules combine two parents to from children for the next generation.

            . Mutation rules apply random changes to individual parents to from children.

  PROCEDURE:-

 1. intitially salalagmite function is defined in a seperate script.

 2.  A negative sign is used in fornt of the function to obtain maxima of the function rather minima.

 3. A neseted for loop is utilzed to iterate each of the elements.

 4.GA is used to determine the optimal of the function

 5. plot is generated for different cases.

 The three cases considered are:

 1. unbounded inputs

 2. Bounded inputs - restricated with upper and lower bounds

 3. Bounded inputs with options - increasing no of iteratons by giving population size.

  syntax:_- 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').

      
  Example:-

   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.)

 Stalagmite Function

 .A Stalagmite function which is represented by these formulas & equations is used to obtain a stalagmite structure

 1). This is then ran through the genetic algorithm command in MATLAB to optimizalution and find more fit value hrough random selection.

  I. The above function defines the stalagmite function which will be used in the main function.

  ii. The input vectors 'x' and 'y' will be assigned with the required values in the main function.

  iii.Negative sign used in the last line is used to maximize the stalagmite function.

• The initial code for generating stalagmite function through GA is :- 

       It gives the output as :-

  .The code runs the stalagmite function through the GA and produces results for minimum value , for global maxima we need the global optimization tool box by matlab .but a simple trick is to put minus sign before the equation output value which gives the negative value, which is actually opposite of minimum and i.e the global maxima of the stalagmite function as shown above. 

  .But the genetic algorithm in this case produces random out every time its made to run .hence to optimize it further several studies are done further to get a more fit value

  Studies for optimization of stalagmite function to find Global Maxima :- 

clear all
close all
clc

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



%creating two dimensional mesh
[xx,yy]=meshgrid(x,y);

num_cases=50
%evaluting 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) = get_stalagmite(input_vector);
end
end
   
tic
%study 1 = satistical behaviour


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

study1_time= toc
%ploting
figure(1)
subplot(2,1,1)
hold on
surfc(xx, yy, f)
shading interp
hold on
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('unbonded inputs')
subplot(2,1,2)
plot(fopt)
xlabel('iterations')
ylabel('function minimum')

tic
% study 2_statistical behavior_with upper and lower bounds
for i = 1:num_cases
[inputs,fopt(1)] =  ga(@get_stalagmite,2,[],[],[],[],[0;0],[0.6;0.6]);
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 minimum')

%study 3_increasing GA iterations
options = optimoptions('ga')
options = optimoptions(options,'populationsize',180)

tic
for i=1:num_cases
[inputs,fopt(i)]=ga(@get_stalagmite,2,[],[],[],[],[0;0],[0.6;0.6],[],[],options);
xopt(i) = inputs(1);
yopt(i) = inputs(2);
end
study3_time = toc
figure(3)
subplot(2,1,1)
hold on
surfc(xx,yy,-f)
hold on
shading interp
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('bounded inpiuts ')
subplot(2,1,2)
plot(-fopt)
xlabel('iteration')
ylabel('function maximum')

 STEP WISE CODE EXPLATION

  .200 cases are assigned to run through the stalagmite function & genetic algorithm 

  .Stalagmite function is called then from the code shown later which produces the stalagmite and is passed to the genetic algorithm in the main command shown above

  .Study 1 : 

  .Is done for 200 iterations as mentioned and further genetic algorithm is used for the stalagmite function comprising of the  equations shown above in the picture with its 2 variables . 

  .The syntax used in this one is :  [inputs , fopt(i)] = ga(@stalagmite, 2);

  .which is : x = ga(fun,nvars) finds a local unconstrained minimum, x, to the objective function, fun. nvars is the dimension number of design variables) of fu

  .The study is timed using tic toc commands 

  .The study 1 is done for statistical behaviour of the stalagmite function and its values are random and changes and not much optimized .

 . markers are plotted using plot3 command over the fitness values obtained in the first study and also a 2d graph is shown for a  more understanding to its global maxima.

 . for first study there is a high variation

 . Study 2 : 

 . This study is done for further optimization of the stalagmite function . 

 . Similar approach as first study is made and the syntax for GA is further modified and upper and lower bounds are assigned from 0-1 for both x and y. 

  [inputs , fopt(i)] = ga(@stalagmite, 2, [],[] ,[],[],[0;0],[1;1]);

 .syntax - x = ga(fun,nvars,A,b,Aeq,beq,lb,ub)  , where A ,B and Aeq and beq are blanks . lb and ub represents lower and upper  bounds respectively. 

 .It 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.)

. The rest approach is same as study 1 .

 .Study 3 :

. Study 2 sucessfully optimized the function but it can still be further optimized, hence we do further one more study where we increase the ga populationsize by modifying the syntax for GA and set it to 500 from default 50

 .[inputs , fopt(i)] = ga(@stalagmite, 2, [], [] ,[],[],[0;0],[1;1],[],[],options); 

 .Syntax - 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.

 .where values not placed are simply replaced with blank brackets .

 .Again similar approach as study 1 is followed.

 .CALLING STALGMITE FUNCTION:-

  . This is stalagmite function developed from the formulas and equations above to produce stalagmite and its used in the GA and it also follows from the get_stalagmite funtion and then passed through studies conducted.

   . For study 1 below it is seen the value varies alot in negative direction . (global maxima)

   . Time study 1

    .For study 2 the value is optimized and the global maxima is bounded between 0-1 and it still needs more optimization and shows the global maxima value ranges between -1 and -0.9 majorly

   . Study time 2.

  3. In study 3 the genetic algorithm has highly optimized the value for 150 iterations and 180 population size .Our global maxima range lies between - 0.96932615 to -0.96932614 even though it still shows slight variation as can be       seen in the 2d  graph. but the stalagmite algorithm is considerably optimize.

  Study time3:-