About the initial population in Genetic Algorithm Toolbox

I have a problem in generating an initial population which meets the constraints in using the Genetic Algorithm Toolbox.

The test problem is as follows:

There are two integral variables x1 and x2. Both of them can only be 0 or 1. The objective function is: z=x1+x2.The constraint is x1+x2<2.

The codes are:

options=gaoptimset('InitialPopulation',[1 0]); A = [ 1 1]; b = 1; lb = zeros(2,1); ub=ones(2,1); intcon=[1 2]; [x,fval,exitflag] = ga(@test,2,A,b,[],[],lb,ub,[],intcon,options);

%******************************************************************** function [y]=test(X) y=-X(1)-X(2); end

To check the initial population generated by the ga toolbox, one line(bold) is added in the file of makeState.m:

% If InitialPopulation is partly empty we will use the creation function to % generate population (CreationFcn can utilize InitialPopulation) if sum(options.PopulationSize) ~= size(options.InitialPopulation,1) % If initial population is empty and we know one feasible individual then % assign it to InitialPopulation if ~isempty(Iterate.x) && isempty(options.InitialPopulation) options.InitialPopulation(1,:) = Iterate.x'; fprintf('makeState: InitialPopulation is not used, Iterate.x\n'); end fprintf('makeState: InitialPopulation is partially used, feval\n'); state.Population = feval(options.CreationFcn,GenomeLength,FitnessFcn,options,options.CreationFcnArgs{:}); assignin('base','InitialPopulationSet',state.Population); % evalin('base','InitialPopulationSet=state.Population');

However, I found a problem with the initial population when x1,x2 are integers(intcon=[1 2]). Some of them do not meet the constraint of x1+x2<2 InitialPopulationSet =

     1     0
     0     0
     0     0
     1     0
  *   1     1*
     1     0
     0     0
  *   1     1*
     0     0
     1     0
     0     1
     1     1
     0     1
     0     1
     0     0
     ......

When I set intcon=[], which means the constrint of integer is released for x1 and x2, all the initial population meet the constraints: InitialPopulationSet =

    1.0000         0
         0         0
    1.0000         0
         0    1.0000
    0.2381    0.7619
    0.5074    0.4926
    0.7263    0.2737
    0.3194         0
         0    0.2737
    1.0000         0
    0.2910    0.3516
    0.8405    0.0000
    0.3612    0.6388
    0.5074    0.4926
    0.0000    0.0000
    ......

Can anybody help me to explain why does this situation happen? If the place where I obtain the initial population is wrong, where can I get it correctly?

Thank you very much！

Tony