I'm trying to solve the minimization problem
$$\inf_{x \in X} x^T(Ax + b) + \sum_{i=1}^n x_i \log(x_i)$$
where $b \in \mathbb{R}^n$, $X \subset \mathbb{R}^n$ is a convex set. And $A$ is a symmetric and positive definite matrix. The optimality conditions give me something like
$$-b \in 2Ax + \vec{1} + L(x) + N_X(x)$$
where $L(x)$ represents the vector in $\mathbb{R}^n$ obtained by taking the elementwise logarithm of $x$; i.e., $[L(x)]_i=\log x_i$.
The problem is how do I take that cone out of the inclusion. If there wasn't that log term it would become a projection on the set $X$. Is there a trick to make this as a projection on $X$ of other vector or something?
I'd appreciate any help or any paper/book reference where there's some information about this kind of problem