Problem with fmincon and many starting values

Hi all, I'm trying to code the following minimization problem using fmincon and many starting values but I get error messages because the objective function has the log function inside. How can I solve this problem? Here the code. Thank you!

    n=2; 
    thetatrue=[1 1 2 2 0.2]; 
    mu = [0 0];
    sd = [1 thetatrue(2*n+1); thetatrue(2*n+1) 1];

    load data      
    X=data(:,1:n); 
    Y=data(:,n+1:size(data,2));  
    A1(:,2)=-X(:,1); 
    A1(:,1)=-1;   
    A2(:,2)=-X(:,2); 
    A2(:,1)=-1;

    W1=(all(bsxfun(@eq,Y,[0 0]),2)); 
    W2=1-W1;   

    cdfun=@(x) mvncdf( [A1*[x(1);x(3)],  A2*[x(2);x(4)] ]...
                          ,mu,[1 x(5); x(5) 1]);    

    options=optimset('Algorithm','interior-...
                      point','Display','iter','MaxIter',10000,'TolX',10^-30,'TolFun',10^-30);

    % Grid of starting values for fmincon
    alpha1=normrnd(thetatrue(1),1,1,5);   %random draws from N(alpha1true,1)
    alpha2=normrnd(thetatrue(2),1,1,5);   %random draws from N(alpha2true,1)
    beta1=normrnd(thetatrue(3),1,1,5);    %random draws from N(beta1true,1)
    beta2=normrnd(thetatrue(4),1,1,5);    %random draws from N(beta2true,1)
    rho=(-0.9:0.1:0.9);

    CC = {alpha1,alpha2,beta1,beta2,rho};
    b = cellfun(@numel,CC);
    n1 = cumsum(b);
    a = fullfact(b);
    idx = bsxfun(@plus,a,[0, n1(1:end-1)]);
    v = cat(2,CC{:});
    theta0grid = v(idx);  %grid of starting values

    L_nstar=zeros(size(theta0grid,1),1); 
    Exitflag=zeros(size(theta0grid,1),1); 

    for j=1:size(theta0grid,1) %for each starting value
    theta0=theta0grid(j,:); 
    [theta,fval,exitflag,output]=...
            fmincon(@(x) log_lik(x,cdfun,W1,W2),theta0,[],[],[],[],[-Inf; -Inf; Inf; -Inf;-0.999],[+Inf; +Inf; +Inf; +Inf; 0.999],[],options);

    L_nstar(j)=-fval;
    Exitflag(j)=exitflag;
    end

    L_nstar=max(L_nstar);

and

function val=log_lik(theta,cdfun,W1,W2)
        z=cdfun(theta);
        val=-sum(W1.*log(z)+W2.*log(1-z));

[MRC]

for example when j=1, I get the following error

Error using barrier (line 22) Objective function is undefined at initial point. Fmincon cannot continue.

Error in fmincon (line 905) [X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN] = barrier(funfcn,X,A,B,Aeq,Beq,l,u,confcn,options.HessFcn, ...

Answer 1

[Matt J]

The sqp algorithm can recover from NaN's and Infs returned by the objective function and constraints.

[MRC]

I get the same error also with sqp

Error using sqpLineSearch (line 22) Objective function is undefined at initial point. Fmincon cannot continue.

Error in fmincon (line 915) [X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN] = sqpLineSearch(funfcn,X,full(A),full(B),full(Aeq),full(Beq), ...

[MRC]

I have read that both interior-point and sqp can recover from Nan's and Inf's, so why is not the case here?

[Matt J]

I think you need a starting point that is feasible, so that they have a foothold in the feasible set.

You should probably take a closer look at your initial values and at the values of mu and sigma to see why you're getting z=0 or 1. Since z=cdfun(theta) is from a Gaussian CDF, z should always assume values strictly between 0 and 1 where log(z) and log(1-z) should not be producing undefined values. Maybe you have overflow, in which case you're choosing highly extreme theta values, or you have sigma=0 somewhere...