Semidefinite matrix as a constraint

Nonconvex
1

Hello,

I am trying to solve an optimization problem with the following objective

A * B + B’ * A - x * x’ > =0

where A is given n x n matrix, B is a decision variable ( it is an n x n matrix) and x is another decision variable of size n x 1. The size of the sum above is an nxn matrix, I want to impose that it is positive semidefinite. The error that I get is the following:

Disciplined convex programming error:
** Only scalar quadratic forms can be specified in CVX**

2

it appears that you are trying to constrain the LHS to be psd,i.e., in spd mode. The inclusion of x*x', with x a decision variable, makes that a BMI (Bilinear Matrix Inequality), which is non-convex, and can’t be handled with CVX.

The only semidefinite constraints which are convex, and which CVX can handle, are affine (linear) in the decision variables.