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Solving Mixed Integer Programs Using Neural Networks

Vinod Nair, Sergey Bartunov, Felix Gimeno, Ingrid von Glehn, Pawel Lichocki, Ivan Lobov, Brendan O’Donoghue, Nicolas Sonnerat, Christian Tjandraatmadja, Pengming Wang, Ravichandra Addanki, Tharindi Hapuarachchi, Thomas Keck, James Keeling, Pushmeet Kohli, Ira Ktena, Yujia Li, Oriol Vinyals, Yori Zwols
DeepMind, Google Research
arXiv:2012.13349 [math.OC], (23 Dec 2020)

@misc{nair2020solving,

   title={Solving Mixed Integer Programs Using Neural Networks},

   author={Vinod Nair and Sergey Bartunov and Felix Gimeno and Ingrid von Glehn and Pawel Lichocki and Ivan Lobov and Brendan O’Donoghue and Nicolas Sonnerat and Christian Tjandraatmadja and Pengming Wang and Ravichandra Addanki and Tharindi Hapuarachchi and Thomas Keck and James Keeling and Pushmeet Kohli and Ira Ktena and Yujia Li and Oriol Vinyals and Yori Zwols},

   year={2020},

   eprint={2012.13349},

   archivePrefix={arXiv},

   primaryClass={math.OC}

}

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Mixed Integer Programming (MIP) solvers rely on an array of sophisticated heuristics developed with decades of research to solve large-scale MIP instances encountered in practice. Machine learning offers to automatically construct better heuristics from data by exploiting shared structure among instances in the data. This paper applies learning to the two key sub-tasks of a MIP solver, generating a high-quality joint variable assignment, and bounding the gap in objective value between that assignment and an optimal one. Our approach constructs two corresponding neural network-based components, Neural Diving and Neural Branching, to use in a base MIP solver such as SCIP. Neural Diving learns a deep neural network to generate multiple partial assignments for its integer variables, and the resulting smaller MIPs for un-assigned variables are solved with SCIP to construct high quality joint assignments. Neural Branching learns a deep neural network to make variable selection decisions in branch-and-bound to bound the objective value gap with a small tree. This is done by imitating a new variant of Full Strong Branching we propose that scales to large instances using GPUs. We evaluate our approach on six diverse real-world datasets, including two Google production datasets and MIPLIB, by training separate neural networks on each. Most instances in all the datasets combined have 10^3−10^6 variables and constraints after presolve, which is significantly larger than previous learning approaches. Comparing solvers with respect to primal-dual gap averaged over a held-out set of instances, the learning-augmented SCIP is 2x to 10x better on all datasets except one on which it is 10^5x better, at large time limits. To the best of our knowledge, ours is the first learning approach to demonstrate such large improvements over SCIP on both large-scale real-world application datasets and MIPLIB.
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