Fminunc returns indefinite Hessian matrix for a convex objective

Question

In minimizing a convex objective function, does it mean that the Hessian matrix at minimizer should be PSD? If fminunc in Matlab returns a hessian which is not psd what does it mean? am I using a wrong objective?

Answer 1

I do that in environments other than matlab.

Non-PSD means you can't take the Cholesky transform of it (i.e. the matrix square-root), so you can't use it to get standard errors, for example.

To get a good hessian, your objective function has to be really smooth, because you're taking a second derivative, which doubly amplifies any noise. If possible, it is best to use analytic derivatives rather than finite-difference. That is, if you really need the hessian.