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Learning directed acyclic graphs (DAGs) from data is a challenging task bothin theory and in practice, because the number of possible DAGs scalessuperexponentially with the number of nodes. In this paper, we study theproblem of learning an optimal DAG from continuous observational data. We castthis problem in the form of a mathematical programming model which cannaturally incorporate a super-structure in order to reduce the set of possiblecandidate DAGs. We use the penalized negative log-likelihood score functionwith both $\ell_0$ and $\ell_1$ regularizations and propose a new mixed-integerquadratic optimization (MIQO) model, referred to as a layered network (LN)formulation. The LN formulation is a compact model, which enjoys as tight anoptimal continuous relaxation value as the stronger but larger formulationsunder a mild condition. Computational results indicate that the proposedformulation outperforms existing mathematical formulations and scales betterthan available algorithms that can solve the same problem with only $\ell_1$regularization. In particular, the LN formulation clearly outperforms existingmethods in terms of computational time needed to find an optimal DAG in thepresence of a sparse super-structure.

Integer Programming for Learning Directed Acyclic Graphs from Continuous Data