general optimization over PSD matrices

[Anil]

Hi all,

 I'm interested in solving the following optimization problem, over the space of PSD matrices. (This question is a bit general, so let me know if a different group is more appropriate for this.)

Let

  G ~ MVN(0, K, U), 

where K is a known NxN PSD matrix, and U is an unknown PxP PSD matrix

  E ~ MVN(0, I, V),

where I is the NxN identity matrix, and V is an unknown PSD matrix.

  Y = G+E

is, thus, the sum of two matrix-variate normal distributed random variables.

Given Y (and K), I wish to iteratively find

   U* = argmax_U log p(Y | U, V*)

   V* = argmax_V log p(Y | U*, V)

I believe that the log likelihood above is concave w.r.t U, and separately, V. How could I proceed in optimizing this function over U (and V), either using cvxopt or other similar software?

While there are simple transformations that allow fast and efficient computation of the likelihood above, the gradient and hessian are fairly costly to compute for even moderately sized problems (as far as I can tell). Could you provide suggestions on how to approximate the gradient and hessian numerically?

thanks much!

cheers

Anil