Convex optimization with positive definite and symmetry constraints

Convex optimization with positive definite and symmetry constraints

I have this function $$f(X) = -\frac{1}{2}y^T\mathrm{X} y +
b^T\mathrm{X}^{-1}b + \frac{1}{2}\log|\mathrm{X}|$$ where $\mathrm{X}$ is
a matrix consisting of three components $\mathrm{X = A+B+C}$ where
$\mathrm{A}$ is a diagonal element matrix with $a$ in the diagonal terms,
$\mathrm{B}$ is a diagonal matrix with $b$ in the diagonal terms,
$\mathrm{C}$ is a matrix with elements $c_{ij}$. I want to optimize over
the $c_{ij}$ the above function or lets say matrix $\mathrm{C}$. So what I
can do is assume $\mathrm{A}$ and $\mathrm{B}$ to be constants and
optimize over $\mathrm{C}$.
How can I do it with the constraints that $\mathrm{C}$ is positive
definite and symmetric. Suggestions?