# A convex programming problem involving sum of logarithms of linear functions

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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This is probably not news to you, but it helped me visualize the problem: We are given $m$ different affine functions on the $(n-1)$-simplex, and we want to maximize (the log of) their product. –  Rahul Jun 15 '12 at 3:45
Isn't this simply the logarithmic barrier? Check section 8.5.3 of Boyd and Vandenberghe. –  TenaliRaman Jun 15 '12 at 3:47
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 vast –  mirror2image Jul 11 '12 at 5:59