Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 205 / 2 / b / Solution

The differential of the log-determinant is $d\log \det \Omega =\mathrm{Tr}({\Omega }^{-1}d\Omega )$. The subdifferential of the entrywise ${\ell }^1$ norm consists of symmetric matrices $Z$ withThe Karush-Kuhn-Tucker conditions for the Graphical Lasso are thereforeBecause $-\log \det \Omega$ is strictly convex on the positive-definite matrices, these conditions characterize the unique minimizer whenever it exists.