Parameter Estimation for Multivariate Normal Distribution

I’m trying to solve a problem of the following form:

\min_\Sigma \quad \log\det(\Sigma) + v^T \Sigma^{-1} v
s.t. \quad \Sigma is (symmetric) positive semidefinite

I tried using log_det and matrix_frac, but it doesn’t work:

Disciplined convex programming error:
   Illegal operation: {concave} + {convex}

Any ideas? Thanks!

(In this example we have closed-form solution, but it’s just a simplified version of my problem.)

Look at exercise 7.4 on pp. 393-394 of Boyd and Vandenberghe “Convex Optimization”

Thank you for you help!

Are you suggesting that adding a constraint \Sigma \preceq 2 v v^T will get the code pass DCP ruleset?

The objective function would not satisfy CVX’s DCP rules even though the optimization problem would be convex. I don’t know how to formulate that problem in CVX, although I am not saying it definitely can not be done.

I don’t know how CVX works, but I’m wondering if I can turn off the convexity check.

You can not turn off CVX"s convexity check.

If you do figure out how to get this accepted by CVX, please let me know, because I am interested in exactly this situation: maximum likelihood estimation of Multivariate Normal parameters subject to additional convex constraints. It would have to be a clever reformulation to get it accepted by CVX.

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