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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.

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    Not a logarithmic barrier because hyperplanes $A_i$ are not active because of simplex constraint. As Rahul said, logarithms could be excessive here, because the problem is the same as maximizing polynomial (product of hyperplanes) on simplex. Poster should check polynomial optimization literature - it's vast2012-07-11

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