Existence of the solution to the graphical lasso

The graphical lasso (glasso) is an $l_1$ penalised likelihood estimator for a Gaussian precision matrix. A benefit of the glasso is that it exists even when the sample covariance matrix is not positive definite but only positive semidefinite. This note collects a number of results concerning the existence of the glasso both when the penalty is applied to all entries of the precision matrix and when the penalty is only applied to the off-diagonals. New proofs are provided for these results which give insight into how the $l_1$ penalty achieves these existence properties. These proofs extend to a much larger class of penalty functions allowing one to easily determine if new penalised likelihood estimates exist for positive semidefinite sample covariance.