How to solve convex >=0 constraints?

Nonconvex
1

Hello, everyone,

My problem is a quadratically constrained quadratic program (QCQP) and can be find as follows:

min V’ * P * V

s.t. sum( V’ * A_i * V ) >=0, i=1,…,m (1)

sum( V’ * B_j * V ) <=0, j = 1,…n

Trace( V * V’ ) >0

where P,A_i,B_j are positive semidefinite. V is a N * 1 vector.

My question is that I can use SDP method to solve this problem and get Q=V * V’, and then decompose Q to get x in order to get the mmse receiving beamformer Ummse. However, the problem is that Q are often full rank according to my simulation results via with the aid of cvx tools. This will conflict with Rank(Q)=1.

If there is no (1), I can solve this QCQP via cvx instead of SDP method . However, since the first constraint is a convex >=0, perhaps the problem above is nonconvex one. How to relax it or deal with it in that case?

I will be appreciated if anyone can transform it to a convex problem which can be solved via the aid of cvx tools.

Look forward to your reply!

Jules

Edit by MLS: Corrected typo in thread title from “concave” to “convex”.

2

Look at stephen boyd’s answer at How to handle nonlinear equality constraints?#35 . Despite the thread name, that response addresses inequality constraints “going the wrong direction”. I offer no opinion as to whether that is the best way, or even a good way in your case.

3

This simply isn’t the right forum for this question. CVX is for models that you already know are convex. You know this one is not. It is outside of the scope of this forum to discuss non-convex models that might be convexified. (Mark’s answer is a pleasant exception.)