help！My optimization problem has hundreds of matrix variables and thousands of constraints, how do I express them effectively?
You can declare a 6D array, in which the last 4 dimensions are over the l indices.
variable G(4,4,4,4,4,4) hermitian
which makes G(:,:,l1,l2,l3,l4) a 4 by 4 hermitian matrix for each value of l1,l2,l3;l4.
Use a 4-fold nested fro loop for the semidefinte constraints. I don’t know of a way
to vectorize that. I think the double sum in the semidefinite constraints can be vectorized - see How to vectorize most constraint loops in CVX by @jackfsuia
The objective can be written as -sum(sum(eta.*p)) or as -sum(eta(:).*p(:)) .
It might take a while for CVX to process the problem. I have no idea how difficult or time consuming it will be for the solver. I suggest using Mosek as solver if it is available to you.
Perhaps you can try a smaller problem first (for instance, the l from 1 to 2, rather than 1 to 4), to see how long that takes for CVX to generate.
Anyone else should feel free to provide other, hopefully better, suggestions.
Ok, I’ll give it a try as you suggested. Thank you for your advice！