How to find a good reparametrization to move from a constraint optimization to an unconstrained one?

Dear all,

I have a little optimization problem. I have decided to move from a constrain optimization to an unconstrained one.

I want to maximise a likelihood function of Markov-Switching GARCH model. The constraints are a little bit complicated.

• All the parameters have to be greater than 0 - Not a problem
• The columns of the transition probability matrix have to sum to one - Not a problem
• And the last, for the stationarity this one: max(abs(eig(Mc))) > 1 with Mc a big matrix containing some elements which depend on the transition matrix and GARCH parameters - Warning: problem

How is it possible to check that?

For the moment, the only solution i could build, is to check after the estimation if the stationarity constraint is verified. If it is not, I would make another estimation with other single constraints on the coefficients. What do you think about that?

Thanks a lot for your help.

Best,

Thomas

Answer 1

[Alan Weiss]

You can indeed incorporate this kind of nonlinear constraint into a constraint for fmincon. In fact, there is an example in the documentation that uses the eigenvalues of a matrix as a goal for fgoalattain. In that example there is the line

 eigfun = @(K) sort(eig(A+B*K*C));

You could make a nonlinear constraint function along the following lines:

 function [c,ceq] = eigfun2(x)
 ceq = [];
 % Put your calculation of Mc here, and then:
 c = 1 - max(abs(eig(Mc)));

If you calculate Mc as part of other things, pass it in a nested function, as explained here.

Alan Weiss

MATLAB mathematical toolbox documentation