I have a convex optimization problem as follows:
\begin{align*} maximize_{x\in R^n} &\sum_{i=1}^n a_i log(x_i)\\ st\quad & \sum_{i=1}^n p_{ij} (x_i-1) = 0 \quad \forall j\\ & x_i \geq 0 \end{align*} where, $a_i \geq 0$ and $P$ is an $n\times m$ probability matrix with rows sum up to 1. Is there any efficient and fast algorithm to solve it? (I suppose it doesn't have a closed-form solution)