# fmincon nonlinear inequality constraint

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hey yo on 9 Aug 2022
Commented: hey yo on 9 Aug 2022
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.

Matt J on 9 Aug 2022
Edited: Matt J on 9 Aug 2022
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 on 9 Aug 2022
Okay, thank you! This is excellent!

### More Answers (1)

Torsten on 9 Aug 2022
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.
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hey yo on 9 Aug 2022
Thank you! I will try to do that artificial adjustment as you suggested.