# An efficient algorithm to solve this convex optimization problem

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)

• Yes, I had included the non-negativity constraint. – ie86 Oct 19 '16 at 15:33

For the equality constraints, you can either work with Lagrange multipliers, or you can eliminate the constraint by only working on the null space of the matrix $P$, assuming that it is easy to compute.