Question

Write a MATLAB script file that will find the values of X and Y required to minimize the function f(x,y) = 30x + 60y + 40pi*y subject the constraining function g(x,y) = 2xy + (pi*y^2)/2 - 1600 using the method of the Lagrange Optimization Technique.

Answer 1

PSO IS ONE OF THE LAGRANGE OPTIMIZATION TECHNIQUE : SO;

MAT LAB CODE FOR THIS:

FUNCTION INPUT CODE:

function z=function_please(x,y)
    z= 30x + 60y + 40pi*y;
end

CONSTRAINS INPUT CODE:

function ret=constraints(x,y)
ret = 30x + 60y + 40pi*y;

%Give constrains
c=[];
c(1) = 2xy + (pi*y^2)/2 - 1600;
s=zeros(1,length(c));
for i=1:length(c)
    if c(i)>=0
        s(1,i)=0;
    else
        s(1,i)=1;
    end
end
excess = inf;
ret=ret+excess*sum(s);

ORGINAL CODE:

clc;
clear;
close all;
%% Problem Definition
fun = @(x,y) function_please (x,y);
nVar =2;

VarSize = [1 nVar];

VarMin = -10;

VarMax = 10;

%% Parameters of PSO

MaxIt = 100; % Maximum Number of Iterations

nPop = 100; % Population or Swarm size

w = 1;
wdamp = 0.98;
c1 = 2;
c2 = 2;

%% Initialization

% The Particle Template
empty_particle.Position = [];
empty_particle.Velocity = [];
empty_particle.funval = [];
empty_particle.Best.Position = [];
empty_particle.Best.funval = [];

%Create Population Array
particle = repmat(empty_particle, nPop, 1);

%Intialize Global Best
GlobalBest.funval = inf;

%Initialize Population Members
for i=1:nPop

    %Generate Random Position
    particle(i).Position = unifrnd(VarMin, VarMax, VarSize);
    %Intialize Velocity
    particle(i).Velocity = zeros(VarSize);
  
    % Evaluation
  
   %particle(i).funval = fun( particle(i).Position);
   particle(i).funval = constraints(particle(i).Position);
  
    %Update the Personal Best
    particle(i).Best.Position = particle(i).Position;
    particle(i).Best.funval = particle(i).funval;
  
    %Update Global Best
        if particle(i).Best.funval < GlobalBest.funval
            GlobalBest = particle(i).Best;
        end
  
end

% Array to Hold Best Cost value on Each Iteration
BestCosts = zeros(MaxIt, 1);
%% Main Loop of PSo
    for it=1:MaxIt
        for i=1:nPop
  
            %Update Velocity
            particle(i).Velocity = (w*particle(i).Velocity)...
                +(c1*rand(VarSize).*(particle(i).Best.Position - particle(i).Position))...
                +(c2*rand(VarSize).*(GlobalBest.Position - particle(i).Position));
      
            %Update Position
            particle(i).Position = particle(i).Position + particle(i).Velocity;

      
      
            %particle
            particle(i).Position = max(particle(i).Position, VarMin);
            particle(i).Position = min(particle(i).Position, VarMax);
      
            %Evaluation
            particle(i).funval = constraints(particle(i).Position);
      
            %Update Personal Best
            if particle(i).funval < particle(i).Best.funval;
          
                particle(i).Best.Position = particle(i).Position;
                particle(i).Best.funval = particle(i).funval;
          
                %Update Global Best
                if particle(i).Best.funval < GlobalBest.funval
                    GlobalBest = particle(i).Best;
                    s=particle(i).Best.Position;
                end
            end
        end
  
        %Store The Best Cost Value
        BestCosts(it) = GlobalBest.funval;
  
        %Display Iterantion Information
        disp(['Iteration ' num2str(it) ':Best Cost = ' num2str(BestCosts(it))]);
  
        % Damping Intertia Coeffient
        w = w * wdamp;
    end
disp(GlobalBest);
figure;
semilogy(BestCosts, 'LineWidth', 2);
xlabel('Iteration');
ylabel('Best Cost');