## Learning a Large Neighborhood Search Algorithm for Mixed Integer Programs

##### Abstract

Large Neighborhood Search (LNS) is a combinatorial optimization heuristic that starts with an assignment of values for the variables to be optimized, and iteratively improves it by searching a large neighborhood around the current assignment. In this paper we consider a learning-based LNS approach for mixed integer programs (MIPs). We train a \emph{Neural Diving} model to represent a probability distribution over assignments, which, together with an off-the-shelf MIP solver, generates an initial assignment. Formulating the subsequent search steps as a Markov Decision Process, we train a \emph{Neural Neighborhood Selection} policy to select a search neighborhood at each step, which is searched using a MIP solver to find the next assignment. The policy network is trained using imitation learning. We propose a target policy for imitation that, given enough compute resources, is guaranteed to select the neighborhood containing the optimal next assignment amongst all possible choices for the neighborhood of a specified size. Our approach matches or outperforms all the baselines on five real-world MIP datasets with large-scale instances from diverse applications, including two production applications at Google. It achieves $2\times$ to $37.8\times$ better average primal gap than the best baseline on three of the datasets at large running times.

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