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Distributed Parallel Structure-Aware Presolving for Arrowhead Linear Programs

Nils-Christian Kempke, Stephen J Maher, Daniel Rehfeldt, Ambros Gleixner, Thorsten Koch, Svenja Uslu

TL;DR

This work presents a structure-aware parallel presolve framework specialized to arrowhead linear programs (AHLPs) and designed for high-performance computing (HPC) environments, integrated into the parallel interior point solver PIPS-IPM++ and demonstrates the scalability and effectiveness of this approach on a diverse set of AHLPs.

Abstract

We present a structure-aware parallel presolve framework specialized to arrowhead linear programs (AHLPs) and designed for high-performance computing (HPC) environments, integrated into the parallel interior point solver PIPS-IPM++. Large-scale LPs arising from automated model generation frequently contain redundancies and numerical pathologies that necessitate effective presolve, yet existing presolve techniques are primarily serial or structure-agnostic and can become time-consuming in parallel solution workflows. Within PIPS-IPM++, AHLPs are stored in distributed memory, and our presolve builds on this to apply a highly parallel, distributed presolve across compute nodes while keeping communication overhead low and preserving the underlying arrowhead structure. We demonstrate the scalability and effectiveness of our approach on a diverse set of AHLPs and compare it against state-of-the-art presolve implementations, including PaPILO and the presolve implemented within Gurobi. Even on a single machine, our presolve significantly outperforms PaPILO by a factor of 18 and Gurobi's presolve by a factor of 6 in terms of shifted geometric mean runtime, while reducing the problems by a similar amount to PaPILO. Using a distributed compute environment, we outperform Gurobi's presolve by a factor of 13.

Distributed Parallel Structure-Aware Presolving for Arrowhead Linear Programs

TL;DR

This work presents a structure-aware parallel presolve framework specialized to arrowhead linear programs (AHLPs) and designed for high-performance computing (HPC) environments, integrated into the parallel interior point solver PIPS-IPM++ and demonstrates the scalability and effectiveness of this approach on a diverse set of AHLPs.

Abstract

We present a structure-aware parallel presolve framework specialized to arrowhead linear programs (AHLPs) and designed for high-performance computing (HPC) environments, integrated into the parallel interior point solver PIPS-IPM++. Large-scale LPs arising from automated model generation frequently contain redundancies and numerical pathologies that necessitate effective presolve, yet existing presolve techniques are primarily serial or structure-agnostic and can become time-consuming in parallel solution workflows. Within PIPS-IPM++, AHLPs are stored in distributed memory, and our presolve builds on this to apply a highly parallel, distributed presolve across compute nodes while keeping communication overhead low and preserving the underlying arrowhead structure. We demonstrate the scalability and effectiveness of our approach on a diverse set of AHLPs and compare it against state-of-the-art presolve implementations, including PaPILO and the presolve implemented within Gurobi. Even on a single machine, our presolve significantly outperforms PaPILO by a factor of 18 and Gurobi's presolve by a factor of 6 in terms of shifted geometric mean runtime, while reducing the problems by a similar amount to PaPILO. Using a distributed compute environment, we outperform Gurobi's presolve by a factor of 13.
Paper Structure (27 sections, 22 equations, 4 figures, 4 tables, 2 algorithms)

This paper contains 27 sections, 22 equations, 4 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contribution
  3. Previous proceedings publication
  4. Notation
  5. Message Passing Interface
  6. Outline
  7. Structure-aware presolving techniques
  8. Presolvers in PIPS-IPM++
  9. Distributed Arrowhead presolving
  10. Parallel postsolving
  11. Arrowhead-specific presolving techniques
  12. Permutation Presolver
  13. Linear Dependencies Presolver
  14. Algorithms and software design
  15. Data communication during presolve
  16. ...and 12 more sections

Figures (4)

  • Figure 1: Constraint matrix with the arrowhead structure of a real-world ESOM.
  • Figure 2: Constraint permutations executed by the permutation presolver.
  • Figure 3: Variable permutations executed by the permutation presolver.
  • Figure 4: Presolve scaling behavior of PIPS-IPM++, PaPILO, and Gurobi.