Here is a convex programming problem I encountered while working on an estimation problem for a mixture of multinomial distributions.
We have a matrix $A_{m \times n}$ containing non-negative real numbers.
We seek to maximize
$$ \sum_{i=1}^m \log (\sum_{j=1}^n a_{ij} \theta_j) $$
such that
$$ \sum_{j=1}^n \theta_j = 1, \theta_j\geq 0 $$
I'm not sure if this has been studied extensively before, and I'd like to have some analytic results about this convex programming problem.