Sparsity-driven Aggregation of Mixed Integer Programs
Liding Xu, Gioni Mexi, Ksenia Bestuzheva
TL;DR
This paper addresses strengthening cutting planes in branch-and-cut for MILPs by learning sparse row aggregations that yield strong c-MIR cuts. It recasts aggregation as an $\ell_0$-norm minimization and solves it with a two-stage LP-based approach: an $\ell_1$ (lasso) formulation to promote sparsity, followed by iterative reweighting if needed to further reduce nonzeros. The method unifies normal-constraint aggregation and bound substitution, is implemented in SCIP, and evaluated on MIPLIB 2017; results show limited average improvement but notable gains on hard instances, with qualitatively sparser aggregated rows than the prior MW heuristic. This work suggests a practical path to adapt aggregation to instance difficulty and motivates extensions to mixed-integer nonlinear programming and faster optimization techniques.
Abstract
Cutting planes are crucial for the performance of branch-and-cut algorithms for solving mixed-integer programming (MIP) problems, and linear row aggregation has been successfully applied to better leverage the potential of several major families of MIP cutting planes. This paper formulates the problem of finding good quality aggregations as an $\ell_0$-norm minimization problem and employs a combination of the lasso method and iterative reweighting to efficiently find sparse solutions corresponding to good aggregations. A comparative analysis of the proposed algorithm and the state-of-the-art greedy heuristic approach is presented, showing that the greedy heuristic implements a stepwise selection algorithm for the $\ell_0$-norm minimization problem. Further, we present an example where our approach succeeds, whereas the standard heuristic fails to find an aggregation with desired properties. The algorithm is implemented within the constraint integer programming solver SCIP, and computational experiments on the MIPLIB 2017 benchmark show that although the algorithm leads to slowdowns on relatively ``easier'' instances, our aggregation approach decreases the mean running time on a subset of challenging instances and leads to smaller branch-and-bound trees.