Parameter Estimation for Multivariate Normal Distribution

1

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.)

2

Look at exercise 7.4 on pp. 393-394 of Boyd and Vandenberghe “Convex Optimization” http://stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf

3

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?

4

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.

5

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

6

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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