Is $\log \det \left( I + \frac 1 {\sigma^2} H F \bar H \right)$ concave?

Question

I want to maximize the capacity function $\log \det \left( I + \frac 1 {\sigma^2} H F \bar H \right)$ with respect to $F$, subject to the constraints:

(1) $\operatorname{trace} F \le Pt$

(2) $\operatorname{trace} G F \bar G \le Ith$

such that:

$F$, of size $12 \times 12$, is a variable complex hermitian positive definite matrix

$I$, of size $6 \times 6$, is the identity matrix

$H$, of size $6 \times 12$, is a constant matrix

$G$ of size $3 \times 12$, is a constant matrix

$\sigma, Pt, Ith$ are scalar constants.

Is the given function convex in $F$? If not, how to make it so?

Answer 1

Note your terminology error. You asked if the function was convex--but you gave us an entire optimization model. You should be asking instead if your model is a convex optimization problem, not a convex function. And in fact, your objective function is concave---but it needs to be. So it's all the more important to get that terminology correct.

The logarithm of the determinant of a symmetric matrix is concave. Your objective function is just the composition of the logdet function with an affine function of $F$, therefore it is concave as well. Thus you may maximize it in a convex optimization setting, as you have done. (Equivalently, you may minimize its negative, as its negative is convex.)

Both of your constraints are simple linear inequalities, and therefore convex as well.