Let $\mathbf K = \mathbf \Sigma^{-1}$ and $\mathbf S$ be respectively the precision and sample covariance matrices of a multivariate gaussian distribution. The Graphical Lasso amounts to minimize the following penalized negative log-likelihood $$\operatorname{min}_{\mathbf K} \mathcal J(\mathbf K) = -\operatorname{logdet}(\mathbf K) + \operatorname{trace}(\mathbf S \mathbf K) + \lambda ||\mathbf K||_1.$$ where $\operatorname{logdet}$ is the logarithm of the determinant, $\lambda$ is an hyper-parameter for sparsity and $||\mathbf K||_1$ is the sum of the absolute value of the matrix entries.