fmincon nonlinear inequality constraint

Hello,

I do a maximum likelihood estimation using fmincon. My code requires doing a Choleski decomposition to the matrix: A=[betas(7) betas(9); betas(9) betas(8)]. Choleski decomposition requires the matrix A to be positive definite. Because A is a 2x2 matrix, positive definiteness require two conditions:

1. beta(7)>0
2. beta(7)*beta(8)-beta(9)^2>0

I set the lowerbound of beta(7) to be 0. To satisfy the second nonlinear inequality constraint, I wrote a function

function [c,ceq]=mycons(betas)
    ceq=[];
    c=-(betas(7)*betas(8)-betas(9)^2);
end

Then I run my fmincon function

[b_out,fval] =fmincon(@(betas)mainlf3(betas,numobs, p),betas,[],[],[],[], lb, ub,@(betas)mycons(betas), options1);

However, the code does not satisfy the nonlinear constraint. It gives me the error:

Error using chol

Matrix must be positive definite.

Can anyone see what I'm doing wrong? Thank you.

Answer 1

[Matt J]

The nonlinear constraints are not obeyed at all iterations. At iterations where they are not obeyed, chol() will give you an error for obvious reasons.

I suggest you parametrize directly in terms of the Cholesky decomposition., i.e., Let L be an unknown matrix and set the constraint,

ceq=L*L'-[betas(7) betas(9); betas(9) betas(8)];

[hey yo]

Thank you! This is excellent. But I didn't fully understand how to specify L. Is it a parameter that needs to be estimated within the process at each iteration?

[Matt J]

Yes, L would be composed of unknowns in addition to beta.

L=[L1,0;
   L2,L3];

[hey yo]

Okay, thank you! This is excellent!

Answer 2

[Torsten]

It is not guaranteed that the "test parameters" betas satisfy the constraints in each iteration of the optimization process.

So you should first check for positive definiteness before calling Chol. If not, enlarge the diagonal elements artificially by a certain amount.

[hey yo]

Thank you! I will try to do that artificial adjustment as you suggested.