How can I solve this optimization problem in CVX?

1

I am trying to solve an optimization problem with quadratic constraints of the
form

minimize\sum_{m=1}^MTr[W_m]

s.t. Tr[R_kW_k]-\gamma_k\sum_{l\neq k}^MTr(R_kW_l)>=\gamma_k\sigma^2_k

W_k>=0 , k=1,…,M

where W_i = w_iw_i^T ,w_i is (10,1) complex vectors ,R_k\in R^{10*10}, M=3

I’m trying to solve the problem in CVX!

the code is here:

cvx_begin  quiet
variable W0(10,10) hermitian;  
variable W1(10,10) hermitian;  
variable W2(10,10) hermitian;
minimize  (trace(W0)+trace(W1)+trace(W2)) ;
subject to
    trace(R0*W0) >= gamma_1*sigma^2 + gamma_1 * (trace(R0*W1)+trace(R0*W2));
    trace(R1*W1) >= gamma_2*sigma^2 + gamma_2 * (trace(R1*W0)+trace(R1*W2));
    trace(R2*W2) >= gamma_3*sigma^2 + gamma_3 * (trace(R2*W1)+trace(R2*W0));
    W0 == semidefinite(10);
    W1 == semidefinite(10);
    W2 == semidefinite(10);
cvx_end

W_i = w_iw_i^T ,but the answer is a real matrix,I don’t know why?
Thank you for devoting your time .

2

Note that the rank-1 constraints W_i = w_i w_i^T are non-convex, so you cannot solve this problem with CVX. Additionally, those troublesome constraints appear to be missing from your CVX implementation.

3

Oh my god! Now dropping this constraint W_i = w_iw_i^T ,the remaining problem is convex. but I cound also not minimize the objective function according to the code above! thanks !

4

Joachim is right about the rank-1 constraints. But if you omit them, the result is a convex relaxation of the intended problem, and it may give you results that are useful to you.

The real problem here is that semidefinite(10) is a real semidefinite matrix. What you’re looking for is hermitian_semidefinite(10).

5

Thank mcg and Joachim Dahl for your kind helps! It entirely solved my problem！