The basic PSO algorithm study notes

PSO study notes

1. bionic ideas

Cluster algorithm simulation birds flying foraging behavior of PSO, through collective cooperation among the bird population achieve their goals. Compared to other evolutionary methods, It has simple, easy to program and so on. PSO mathematical description follows:
Initialization group of particles randomly (random solutions). Then find the optimal solution by iteration. In each iteration, the particle by tracking the two "extremes" (Pbest, gbest) to update themselves.
After finding the optimal values of these two particles by the following formula to update their speed and position.
\ [\ begin {array} { rl} {V_ {t, j} (t + 1) = w} & {V_ {t, j} (t) + c_ {1} \ times r \ text {and} ( ) \ times \ left (pbes t_ {i, j} -x_ {t, j} (t) \ right)} \\ {} & {+ c_ {2} \ times rand () \ times \ left (gbes t_ I {.} J {I -x_. J} (T) \ right)} \} End {Array \] (. 1)
\ [X_ {I, J} (. 1 + T) = X_ {I, J} ( T) + V_ {I, J} (T +. 1), J = 1,2, \ ldots, D \] (2)
\ [I = 1,2, \ ldots, M. M \]
M is the group the total number of particles, \ [{I} V_ \] is the velocity of the particles; pbest gbest and are defined as the best location and position of the optimal particle population. rand is a random number between (0,. 1); \ [X_ {I} \] is the current position of the particle. cl and c2 are learning factor, usually take c1 = c2 = 1.4962, w is the inertia weight factor.

PSO Code

%% Problem Definition

CostFunction=@(z) Fun(z);        % Cost Function
%Ufun=@(o) Ufun
nVar=4;            % Number of Decision Variables

VarSize=[1 nVar];   % Size of Decision Variables Matrix

VarMin=0;         % Lower Bound of Variables
VarMax=10;         % Upper Bound of Variables


%% PSO Parameters

MaxIt=500;      % Maximum Number of Iterations

nPop=30;        % Population Size (Swarm Size)

% PSO Parameters
w=1;            % Inertia Weight
wdamp=0.99;     % Inertia Weight Damping Ratio
c1=1.5;         % Personal Learning Coefficient
c2=2.0;         % Global Learning Coefficient

% If you would like to use Constriction Coefficients for PSO,
% uncomment the following block and comment the above set of parameters.

% % Constriction Coefficients
% phi1=2.05;
% phi2=2.05;
% phi=phi1+phi2;
% chi=2/(phi-2+sqrt(phi^2-4*phi));
% w=chi;          % Inertia Weight
% wdamp=1;        % Inertia Weight Damping Ratio
% c1=chi*phi1;    % Personal Learning Coefficient
% c2=chi*phi2;    % Global Learning Coefficient

% Velocity Limits
VelMax=0.1*(VarMax-VarMin);
VelMin=-VelMax;

%% Initialization

empty_particle.Position=[];
empty_particle.Cost=[];
empty_particle.Velocity=[];
empty_particle.Best.Position=[];
empty_particle.Best.Cost=[];

particle=repmat(empty_particle,nPop,1);

GlobalBest.Cost=inf;

for i=1:nPop
    
    % Initialize Position
    particle(i).Position=unifrnd(VarMin,VarMax,VarSize);
    
    % Initialize Velocity
    particle(i).Velocity=zeros(VarSize);
    
    % Evaluation
    particle(i).Cost=CostFunction(particle(i).Position);
    
    % Update Personal Best
    particle(i).Best.Position=particle(i).Position;
    particle(i).Best.Cost=particle(i).Cost;
    
    % Update Global Best
    if particle(i).Best.Cost<GlobalBest.Cost
        
        GlobalBest=particle(i).Best;
        
    end
    
end

BestCost=zeros(MaxIt,1);

%% PSO Main Loop

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);
        
        % Apply Velocity Limits
        particle(i).Velocity = max(particle(i).Velocity,VelMin);
        particle(i).Velocity = min(particle(i).Velocity,VelMax);
        
        % Update Position
        particle(i).Position = particle(i).Position + particle(i).Velocity;
        
        % Velocity Mirror Effect
        IsOutside=(particle(i).Position<VarMin | particle(i).Position>VarMax);
        particle(i).Velocity(IsOutside)=-particle(i).Velocity(IsOutside);
        
        % Apply Position Limits
        particle(i).Position = max(particle(i).Position,VarMin);
        particle(i).Position = min(particle(i).Position,VarMax);
        
        % Evaluation
        particle(i).Cost = CostFunction(particle(i).Position);
        
        % Update Personal Best
        if particle(i).Cost<particle(i).Best.Cost
            
            particle(i).Best.Position=particle(i).Position;
            particle(i).Best.Cost=particle(i).Cost;
            
            % Update Global Best
            if particle(i).Best.Cost<GlobalBest.Cost
                
                GlobalBest=particle(i).Best;
                
            end
            
        end
        
    end
    
    BestCost(it)=GlobalBest.Cost;
    
    disp(['Iteration ' num2str(it) ': Best Cost = ' num2str(BestCost(it))]);
    
    w=w*wdamp;

end

BestSol = GlobalBest;
cg_curve = BestCost;
%% Results

%Draw objective space
subplot(1,2,2);
semilogy((1:50:MaxIt),cg_curve(1:50:MaxIt),'k-d','markersize',10,'Color','g','LineWidth',1.5)
title('Convergence curve','FontSize',16)
xlabel('Iteration','FontSize',16);
ylabel('Best score obtained so far','FontSize',16);
set(gca,'FontSize',16);

hold on
axis tight
grid off
box on
legend('PSO')
set(legend,'FontSize',12);

The following code is the test function, used to test the convergence of the PSO

function z=Fun(u)

% F1 Sphere [-100,100]
%z=sum(u.^2);

%F2 Schwel [-10,10]
 %z=sum(abs(u))+prod(abs(u));

%F3 [-100,100]
%dim=size(u,2);
%z=0;
%for i=1:dim
  %z=z+sum(u(1:i)-1)^2;
 % end


% F4 [-100,100]
%z=max(abs(u));

%F5 Rosenbrock  [-30,30]
%dim = 30;
%z=sum(100*(u(2:dim)-(u(1:dim-1).^2)).^2+(u(1:dim-1)-1).^2);


%F6 [-100,100]
%z=sum(abs((u+.5)).^2);

%F7 [-1.28,1.28]
%dim=size(u,2);
%z=sum([1:dim].*(u.^4))+rand;

%F8 [-500,500]
%z=sum(-u.*sin(sqrt(abs(u))));

%F9 Rastrign [-5.12,5.12]
%dim=size(u,2);
%z=sum(u.^2-10*cos(2*pi.*u))+10*dim;

%F10 Ackley [-32,32]
%dim=size(u,2);
%z=-20*exp(-.2*sqrt(sum(u.^2)/dim))-exp(sum(cos(2*pi.*u))/dim)+20+exp(1);

%F11 Griewank [-600,600]
%dim=size(u,2);
%z=sum(u.^2)/4000-prod(cos(u./sqrt([1:dim])))+1;

%F12  %[-50,50]
%dim=size(u,2);
%z=(pi/dim)*(10*((sin(pi*(1+(u(1)+1)/4)))^2)+sum((((u(1:dim-1)+1)./4).^2).*...
%(1+10.*((sin(pi.*(1+(u(2:dim)+1)./4)))).^2))+((u(dim)+1)/4)^2)+sum(Ufun(u,10,100,4));
%function o=Ufun(u,a,k,m)
%o=k.*((u-a).^m).*(u>a)+k.*((-u-a).^m).*(u<(-a));


%F13  %[-50,50]
%dim=size(u,2);
%z=.1*((sin(3*pi*u(1)))^2+sum((u(1:dim-1)-1).^2.*(1+(sin(3.*pi.*u(2:dim))).^2))+...
%((u(dim)-1)^2)*(1+(sin(2*pi*u(dim)))^2))+sum(Ufun(u,5,100,4));
%function o=Ufun(u,a,k,m)
%o=k.*((u-a).^m).*(u>a)+k.*((-u-a).^m).*(u<(-a));

%F14  [-65.536,-65.536]
%aS=[-32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32;,...
   % -32 -32 -32 -32 -32 -16 -16 -16 -16 -16 0 0 0 0 0 16 16 16 16 16 32 32 32 32 32];
%for j=1:25
  % bS(j)=sum((u'-aS(:,j)).^6);
%end
%z=(1/500+sum(1./([1:25]+bS))).^(-1);

% F15   %[-5,5]
%aK=[.1957 .1947 .1735 .16 .0844 .0627 .0456 .0342 .0323 .0235 .0246];
%bK=[.25 .5 1 2 4 6 8 10 12 14 16];bK=1./bK;
%z=sum((aK-((u(1).*(bK.^2+u(2).*bK))./(bK.^2+u(3).*bK+u(4)))).^2);
%end

%F16    %[-5,5]
  %z=4*(u(1)^2)-2.1*(u(1)^4)+(u(1)^6)/3+u(1)*u(2)-4*(u(2)^2)+4*(u(2)^4);

  % F17   %[-5,5]
%z=(u(2)-(u(1)^2)*5.1/(4*(pi^2))+5/pi*u(1)-6)^2+10*(1-1/(8*pi))*cos(u(1))+10;



% F18    %[-5,5]
%z=(1+(u(1)+u(2)+1)^2*(19-14*u(1)+3*(u(1)^2)-14*u(2)+6*u(1)*u(2)+3*u(2)^2))*...
 % (30+(2*u(1)-3*u(2))^2*(18-32*u(1)+12*(u(1)^2)+48*u(2)-36*u(1)*u(2)+27*(u(2)^2)));


% F19     %[1,3]
%aH=[3 10 30;.1 10 35;3 10 30;.1 10 35];cH=[1 1.2 3 3.2];
%pH=[.3689 .117 .2673;.4699 .4387 .747;.1091 .8732 .5547;.03815 .5743 .8828];
%z=0;
%for i=1:4
    %z=z-cH(i)*exp(-(sum(aH(i,:).*((u-pH(i,:)).^2))));
%end


% F20       %[0,1]
%aH=[10 3 17 3.5 1.7 8;.05 10 17 .1 8 14;3 3.5 1.7 10 17 8;17 8 .05 10 .1 14];
%cH=[1 1.2 3 3.2];
%pH=[.1312 .1696 .5569 .0124 .8283 .5886;.2329 .4135 .8307 .3736 .1004 .9991;...
%.2348 .1415 .3522 .2883 .3047 .6650;.4047 .8828 .8732 .5743 .1091 .0381];
%z=0;
%for i=1:4
 % z=z-cH(i)*exp(-(sum(aH(i,:).*((u-pH(i,:)).^2))));
%end

% F21      %[0,10]
%aSH=[4 4 4 4;1 1 1 1;8 8 8 8;6 6 6 6;3 7 3 7;2 9 2 9;5 5 3 3;8 1 8 1;6 2 6 2;7 3.6 7 3.6];
%cSH=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5];
%z=0;
%for i=1:5
  %z=z-((u-aSH(i,:))*(u-aSH(i,:))'+cSH(i))^(-1);
%end


%F22      %[0,10]
aSH=[4 4 4 4;1 1 1 1;8 8 8 8;6 6 6 6;3 7 3 7;2 9 2 9;5 5 3 3;8 1 8 1;6 2 6 2;7 3.6 7 3.6];
cSH=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5];
z=0;
for i=1:7
    z=z-((u-aSH(i,:))*(u-aSH(i,:))'+cSH(i))^(-1);
end


% F23      %[0,10]
%aSH=[4 4 4 4;1 1 1 1;8 8 8 8;6 6 6 6;3 7 3 7;2 9 2 9;5 5 3 3;8 1 8 1;6 2 6 2;7 3.6 7 3.6];
%cSH=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5];
%z=0;
%for i=1:10
    %z=z-((u-aSH(i,:))*(u-aSH(i,:))'+cSH(i))^(-1);
%end




%F����[-10��10]

%x=u(1,1);  
%y=u(1,2);  
%temp=x^2+y^2;  
%z=0.5+(sin(sqrt(temp))^2-0.5)/(1+0.001*temp)^2;
%for i=1:30
  % dim=size(u,2);
   %temp= i-1/dim-1;
   %  z= sum((10.^(6*temp)).* u.^2);      
    
%end   

notes:

% This symbol indicates a comment in matlab means shortcut is ctrl + r, uncommented is ctrl + l, need to use the test function on the abolition of good comments