Given the symmetric positive definite matrix $W\in\mathbb{R}^{n,n}$ and the positive scalar $\lambda $, the objective reads $$\min_{x} x^{T}Wx + \lambda \|\log(x)\|_1, $$ where $\|\cdot\|_1$ denotes the $\ell_1$ norm of the argument and the operator $\log(\cdot)$ is component-wise.
How can I obtain the minimum point of this objective? Does the closed form solution exists if $W\neq I_n$?
In addition, how about the general method to solve it if the $\ell_1$ norm is replaced by some other non-smooth functions?