I want to maximize this capacity function:

maximize(log_det(I + inv(sigma^2.*I)HF*H’)) subject to the constrains:

(1) trace(F) <= Pt should be a positive value

(2) trace(GFG’) <= Ith should be a positive value

such that:

(’): is the complex conjugate of a matrix

F : 12x12 is a variable complex hermitian positive definite matrix

I : is a 6x6 identity matrix

H : 6x12 is a constant matrix

G : 3x12 is a constant matrix

Pt: is a scalar constant

Ith: is a scalar constant

sigma: is a scalar constant

H & G were just randn complex matrices

the code is

Nr = 6;

Nt = 10;

Np = 3;

H = randn(Nr,Nt)+randn(Nr,Nt);

G = randn(Np,Nt)+randn(Np,Nt);

I = eye(Nr,Nr);

Pt = 10;

Ith = 1;

sigma = 5;

cvx_begin sdp

variable F(Nt,Nt) complex hermitian semidefinite

% variable F(Nt,Nt) symmetric semidefinite

minimize(-det_rootn(I+(5^-2.*I) HF*H’))

% minimize(-log_det(I+(5^-2.*I) HF*H’))

subject to

trace(G*F*G’) >=0;

trace(F) >=0;

trace(F) <= Pt;

trace(G*F*G’) <= Ith;

cvx_end

result:

Status: Unbounded

Optimal value (cvx_optval): -Inf