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Sparsity-Driven Aggregation of Mixed Integer Programs

Authors Liding Xu , Gioni Mexi , Ksenia Bestuzheva

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Solvers
  • Mathematics of computing → Mixed discrete-continuous optimization
  • Mathematics of computing → Linear programming
Keywords
  • mixed integer linear programming
  • cutting plane
  • valid inequality
  • separation
  • aggregation
  • projection
  • sparse optimization

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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 𝓁₀-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 𝓁₀-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.

Cite As Get BibTex

Liding Xu, Gioni Mexi, and Ksenia Bestuzheva. Sparsity-Driven Aggregation of Mixed Integer Programs. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SEA.2025.27

Author Details

Liding Xu
  • Zuse Institute Berlin, Germany
Gioni Mexi
  • Zuse Institute Berlin, Germany
Ksenia Bestuzheva
  • Zuse Institute Berlin, Germany

Funding

The work for this article has been conducted in the Research Campus MODAL funded by the German Federal Ministry of Education and Research (BMBF) (fund numbers 05M14ZAM, 05M20ZBM).

Acknowledgements

We want to thank Felipe Serrano for his insightful comments on this paper.

Supplementary Materials

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