DCP Reformulation

To ensure convexity, the following optimization problem must adhere to the rule of perspective of a function:

A = rand(3,3) + 1i * rand(3,3);
A = A + A’;
A = A * A’; % to ensure A being hermitian semidefinite

variables t mu_3
variable W(3,3) hermitian semidefinite
minimize t
subject to
trace(A*W)/(t-mu_3) + trace(W)/mu_3 <= 1;
trace(W) <= 1;
t >= mu_3;
mu_3 >= 0;

However, I’m uncertain about how to reformulate the problem to comply with the DCP rules. If anyone could assist me, I would greatly appreciate it.

Please show us the proof that the first constraint is convex.

According to “the perspective of a function” rule in Boyd’s book,

  1. trace(AW) is convex and 1/(t-mu_3) is positive. Hence, trace(AW)/(t-mu_3) is convex.
  2. trace(W) is convex and 1/mu_3 is positive. Hence, trace(W)/mu_3 is convex.

Therefore the sum of 1 and 2 is convex and having the new convex function smaller than a constant scalar is convex. Is it right?

Oh, sorry. I’m totally wrong!

Yes, you are totally wrong.