Week 4.1 - Genetic Algorithm

Syntax of 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 function.

• 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 Cand 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)

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(___)

ga(___), for any previous input arguments, also returns fval, the value of the fitness function at x.

• [x,fval,exitflag,output] = ga(___)

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(___)

ga(___) also returns a matrix population, whose rows are the final population, and a vector scores, the scores of the final population.

Genetic algorithm Program

• Stating the inputs for search the space x, y with the help of linspace command.
• Using [xx, yy] = meshgrid(x, y), to return 2-D grid coordinates based on the coordinated contained in vector x and y. Xis a matrix where each row is a copy of x, and Y is a matrix where each column is a copy of y. The grid represented by the coordinates X And Y has length(y) rows and length(x) columns.

Evaluating the function

%write a program stalaqmite fuction
function [f] = stalagmite(input_vector)

%Stating the inputs
x = input_vector(1); 
y = input_vector(2);

%Equation of f1x, f1y, f2x, f2y
f1 = (sin(5.1*pi*x + 0.5))^6;
f2 = (sin(5.1*pi*y +0.5))^6;
f3 = exp(-4*log(2)*((x-0.0667)^2/0.64));
f4 = exp(-4*log(2)*((y-0.0667)^2/0.64));

%Fuction as per formula given
f = -(f1*f2*f3*f4);
end

• Plotting the 3D Surface with respect to x ,y and f values with the command of surfc
• Genetic algorithm has been executed with command of ga
• The program is as follows

  clear all
  close all
  clc
  
  % Define Search Space
  x = linspace(0,0.6,150);
  y = linspace(0,0.6,150);
  
  % Creating two dimensional mesh
  [xx,yy] = meshgrid(x,y);       
  
  % EVALUATING THE STALGAMITE FUNCTION
  for i=1: length (xx)                        %For loop for x values
      for j=1: length(yy)                     %For loop for x values
          input (1) = xx(i,j);         %the xx valve of the same is defined as input vector 1
          input (2) = yy(i,j);         %the xx valve of the same is defined as input vector 2
          f(i,j) = stalagmite(input); % Staglagmite function is called (Staglagmite is stucture shows as peaks) 
      end
  end
  
  % CREATING 3D SURFACE OF THE STALAGMITE FUNCTION
  surfc(xx, yy, f)   
  xlabel('X-value')
  ylabel('Y- value')
  shading interp
  
  % RUNNING THE GA ALGORITHM
  [inputs, fval] = ga(@stalagmite, 2)​

  Study of Global Minima using GA

  Study 1: Evaluating GA without bounds

  • The function is iterated 50 times as the given.
  • Output are stored in row vector ‘input_vector’ and function valve is stored in array ‘fopt’
  • The time taken by this function is calculated by tic and toc comment.
  • Using Surfc() and subplot comment the graph is produced for valuation
  • With this global minima cannot be find out.

  Study 2: Evaluating GA with upper and lower bonds

  • As same as the previous study, we here introduce the Upper and lower bounds
  • x = ga(fun,nvars,A,b,Aeq,beq,lb,ub) took this syntax and gave valve for lower bound and upper bound so that the search can be minimalized
  • Plot for the same is done

  Study 3: Evaluating GA by increasing the population

  • Population option is let you to describe the require population that the genetic algorithm uses.
  • Options = optimoptions (solvername, option), where solvername is GA and option to increase the PopulationSize.
  • ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options) with this syntax the optimization is done by GA

  Program is as follows

  clear all
  close all
  clc
  
  % Define Search Space
  x = linspace(0,0.6,150);
  y = linspace(0,0.6,150);
  numcase = 50;
  
  % Creating two dimensional mesh
  [xx,yy] = meshgrid(x,y);       
  
  % EVALUATING THE STALGAMITE FUNCTION
  for i=1:length(xx)                         %For loop for x values
      for j=1:length(yy)                     %For loop for y values
          input_vector (1) = xx(i,j);        %the xx valve of the same is defined as input vector 1
          input_vector (2) = yy(i,j);        %the yy valve of the same is defined as input vector 2
          f(i,j) = stalagmite(input_vector); % Staglagmite function is called (Staglagmite is stucture shows as peaks) 
      end
  end
  
  % Evaluating GA without bounds
  tic
  for k=1:numcase
      [inputs,fopt(k)] = ga(@stalagmite,2);
      xopt(k) = inputs (1);
      yopt(k) = inputs (2);
  end
  study1_time = toc
  
  % Evaluating GA with lower and upper Bounds
  tic
  for k=1:numcase
      [inputs1,fopt1(k)] = ga(@stalagmite,2,[],[],[],[],[0;0],[0.6;0.6]);
      xopt1(k) = inputs1 (1);
      yopt1(k) = inputs1 (2);
  end
  study2_time = toc
  
  % Evaluating GA by Increasing the population
  options = optimoptions('ga','PopulationSize',200);
  
  tic
  for k=1:numcase
      [inputs2,fopt2(k)] = ga(@stalagmite,2,[],[],[],[],[0;0],[0.6;0.6],[],[],options);
      xopt2(k) = inputs2 (1);
      yopt2(k) = inputs2 (2);
  end
  study3_time = toc
  
  
  %Plotting for Studies without bounds
  figure(1)
  subplot(2,1,1)
  hold on
  surfc(xx,yy,f)
  shading interp
  title ('Studies without bounds')
  plot3(xopt,yopt,fopt,'marker','o','markersize',5,'markerfacecolor','r')
  subplot (2,1,2)
  plot (fopt)
  xlabel ('iternation')
  ylabel ('function minima')
  
  %Plotting for Studies with lower and upper Bounds
  figure (2)
  subplot(2,1,1)
  hold on
  surfc(xx,yy,f)
  shading interp
  title ('Studies with lower and upper Bounds')
  plot3(xopt1,yopt1,fopt1,'marker','o','markersize',5,'markerfacecolor','r')
  subplot (2,1,2)
  plot (fopt1)
  xlabel ('iternation')
  ylabel ('function minima')
  
  %Plotting after Increasing the population
  figure (3)
  subplot(2,1,1)
  hold on
  surfc(xx,yy,f)
  shading interp
  title ('By Increasing the population')
  plot3(xopt2,yopt2,fopt2,'marker','o','markersize',5,'markerfacecolor','r')
  subplot (2,1,2)
  plot (fopt2)
  xlabel ('iternation')
  ylabel ('function minima')
  
  

  Results :

• Above program Results shows various optimization to find global minima using GA
• As the results shows the timing for each studies from the by increasing the population time taken will also increases
• the time taken for each studies is as follows
• 
• The Graph shows the that best way to find the global minima will be by increasing the population
• the graphs are attached bellow