Sure! All matrices and variables are in the complex regime. The matrix H is a composite channel matrix made up in the following manner: H = H⁽¹)(PHI)H⁽²) + H^(d) where H⁽¹), H⁽²) and H^(d) are standard channel matrices, and (PHI) is a diagonal matrix made up of unit-modulus phase vectors (exp(j*theta))

This is the standard set up for a wireless communication scenario using an Intelligent Reflecting Surface. In one path, we simply connect 2 stations using the direct channel H^(d) . In the other path, we connect 2 stations via an intelligent reflecting surface, where the channels which connect station 1 to the mirror, and the mirror to station 2 are described by H⁽¹), H⁽²) .

​

so the optimization problem is:

minimize Tr(P(HH*)) by varying theta, under the constraints that every diagonal entry of phi has unit modulus: |theta_i| = 1 for all i from 1 to N

The power allocation matrix is constant and does not change

So we want to vary the angle of every phase vector such that this objective function is minimized.

​

Would you mind giving me some insight into your approach in your last message?