I want to solve the following optimization problem

$\text{maximize}f(X)=-\mathrm{log}\mathrm{d}\mathrm{e}\mathrm{t}(X+Y)-{a}^{T}(X+Y{)}^{-1}a\phantom{\rule{0ex}{0ex}}\text{subject to}X\u2ab0W,$

where the design variable $X$ is symmetric positive semidefinite, $W,Y$ are fixed symmetric positive semidefinite matrices, and $a$ is a given vector. The inequality $X\u2ab0W$ means that $X-W$ is symmetric positive semidefinite.

I was wondering if there's hope to find an analytical solution to either the constrained or unconstrained problem. And if there is none, could I use convex optimization techniques to solve this numerically?

$\text{maximize}f(X)=-\mathrm{log}\mathrm{d}\mathrm{e}\mathrm{t}(X+Y)-{a}^{T}(X+Y{)}^{-1}a\phantom{\rule{0ex}{0ex}}\text{subject to}X\u2ab0W,$

where the design variable $X$ is symmetric positive semidefinite, $W,Y$ are fixed symmetric positive semidefinite matrices, and $a$ is a given vector. The inequality $X\u2ab0W$ means that $X-W$ is symmetric positive semidefinite.

I was wondering if there's hope to find an analytical solution to either the constrained or unconstrained problem. And if there is none, could I use convex optimization techniques to solve this numerically?