Sparsity-driven Aggregation of Mixed Integer Programs

TL;DR

This paper introduces a sparsity-driven method for aggregating constraints in mixed-integer programming, improving solver efficiency on difficult instances by reducing branch-and-bound tree size.

Contribution

It formulates the aggregation problem as an $ ext{l}_0$-norm minimization and employs a novel combination of lasso and reweighting techniques for sparse solutions.

Findings

Reduces mean solving time on challenging MIP instances.

Produces smaller branch-and-bound trees.

Outperforms greedy heuristics in certain cases.

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.