# Matrix Maximization.

Hi everyone,

I'm trying to solve/define the following optimization problem: $\max_M f(M)$ s.t. $M \theta = b$ $\sum_j \theta_j = 1$

When: M is an m*n real matrix $\theta$ and $b$ are n*1 column vectors. f returns a real scalar.

I cannot seem to recall, find a reference, or even define the name of this problem.

What are standard regularity conditions imposed (in terms of $f()$ and $M$) for a solution to exist? What would be the relevant first and second order conditions?

Thanks!

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You could start by using Lagrange multipliers. – Hans Engler Jul 31 '12 at 13:52
That was my guess. My main question is whether you write an FOC for each element of the matrix, and then extra $n$ FOCs for the multipliers resulting from the contraints, or whether the matrix FOCs can somehow be applied for vectors, as the columns are constrained and not really free to move independently. I guess I'll go the element by element way. – Ron Aug 1 '12 at 5:54
@Ron: you might have better luck with this question at math.stackexchange.com --- essentially, you have $m\times n$ variables, so the brute force way is to write FONC for each element. Using some matrix differential calculus, depending on $f$, you might be able to write these conditions more compactly... – Suvrit Aug 1 '12 at 6:53