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Consider the following convex problem $\max_{\mathbf{A}} \log |\mathbf{I} + \mathbf{CA} |$ subject to $\log |\mathbf{I} + \mathbf{FA} | \leq \xi$ where $\mathbf{A}$ is a semidefinite positive matrix and $\mathbf{I}$ is the identity matrix.

Does exist any closed form solution to $\mathbf{A}$? If not, does exist efficient algorithms to solve this problem?

Thanks in advance,

Mikitov

flag	asked Dec 25 2010 at 19:00 mikitov 105●7

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I am curious as to whether the problem is really convex. The determinant is log-concave for positive definite arguments, and maximization of a concave function is equivalent to the minimization of a convex function -- so that's ok; however, your constraint is concave. Can you tell us anything about $\mathbf{C}$ and $\mathbf{F}$? – Gilead Dec 25 2010 at 19:19

I'm sorry, I meant your constraint may be concave, not is concave. – Gilead Dec 25 2010 at 19:34

Also, I think it is possible that the problem will not have a simple closed-form solution. If you consider the KKT system for this problem, the inequality constraint will induce complementarity constraints ($\mu \cdot (\log |\mathbf{I} + \mathbf{FA} | - \xi) =0$), which need to be resolved algorithmically. – Gilead Dec 26 2010 at 2:32

$\mathbf{F}$ and $\mathbf{C}$ are meant to be herminitian. Yes, being rigourous the optimization problem is $\min_{A} \log | \mathbf{I} + \mathbf{FA}|$ subject to $\log | \mathbf{I} + \mathbf{CA}| \geq \epsilon$ and also $\mathbf{A}$ being semidefinite positive. Can this problem be casted as a convex problem? Ok about the closed form solution. Thanks, Mikitov – mikitov Dec 26 2010 at 10:31

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Off-hand, it seems to me that the problem is convex only if $\mathbf{I}+\mathbf{CA}\succ\mathbf{0}$ and $\mathbf{I}+\mathbf{FA}\prec\mathbf{0}$. If that is the case, you may be able to pose this as some kind of semidefinite program -- but that takes me out of my province. – Gilead Dec 26 2010 at 22:00

logdet maximization under logdet inequalitie constraint

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