Symmetric positive semidefinition fixed rank optimization

[Prokins Wang]

Hi everyone,

I find the manifold "symfixedrankYYcomplexfactory(n, k)" very powerful. I'm trying to find a real number x such that the matrix G−xB satisfies the symmetric positive semidefinite and fixed-rank constraints, i.e.,

where G and B are given positive semidefinite matrices.

The optimization variable is x, but the constraint is imposed on the matrix G-x*B. How should I utilize the manifold "symfixedrankYYcomplexfactory(n, k)"? Any comments would be appreciated!

Best

Prokins

[Nicolas Boumal]

Hello,

Since the variable is x and that is a real number, the problem as stated would not fit as an optimization problem on symfixedrankYYcomplexfactory.

You might reformulate it by introducing a new variable Z and optimize this:

min || G - xB - Z ||^2

for the variables x in R and Z in symfixedrankYYcomplexfactory.

If somehow you can minimize this down to 0, then indeed G - xB = Z, and Z satisfies the constraints, so you would be fine.

I don't know if that's the best approach though. This is really a one-dimensional search problem, with quite a bit of structure. Also, the matrix G - xB is the form that comes up naturally when considering the matrix pencil of (G, B). Since you want G-xB to have rank k (presumably that's < n), then we know det(G - xB) = 0. In other words, x must be an "eigenvalue" for the pencil (G, B). In Matlab, you can compute those with eig(G, B). Then, you are left with a finite number of candidate values for x that you can try one by one. If none of those succeed, then you know there is no solution.

Best,

Nicolas

[Prokins Wang]

Dear Professor Nicolas,

         Thanks for your kind reply! It helped me a lot!

Best

Prokins