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Error Bounds for Rank-one Double Nonnegative Reformulations of QAP and Exact Penalties

Yitian Qian, Shaohua Pan, Shujun Bi, Houduo Qi

TL;DR

This work addresses the quadratic assignment problem by focusing on three equivalent rank-one doubly nonnegative reformulations. It establishes locally Lipschitz error bounds for the associated feasible set and proves global exact penalties for both the DC reformulation and BM factorizations, enabling a practical relaxation approach. An ALM-based relaxation method, EPalm, is developed to search for rank-one approximate feasible solutions, and numerical experiments on QAPLIB and Drezner instances show EPalm often delivers superior objective values compared with Gurobi within the same time budget. The results remove the need for calmness assumptions in prior penalty analyses and provide a scalable path toward high-quality feasible solutions for large QAP instances, with clear directions for future improvement in scalability and bound refinement.

Abstract

This paper focuses on the error bounds for several equivalent rank-one doubly nonnegative (DNN) conic reformulations of the quadratic assignment problem (QAP), a class of challenging combinatorial optimization problems. We provide three equivalent rank-one DNN reformulations of the QAP, including the one proposed in \cite{Jiang21}, and establish the locally and globally Lipschitzian error bounds for their feasible sets. Then, these error bounds are employed to prove that the penalty problems induced by the difference-of-convexity (DC) reformulation of the rank-one constraint are global exact penalties, and so are the penalty problems for their Burer-Monteiro (BM) factorizations. As a byproduct, the penalty problem for the rank-one DNN reformulation in \cite{Jiang21} is shown to be a global exact penalty without the calmness assumption. Finally, we illustrate the application of these exact penalties by proposing a relaxation approach with one of them to seek a rank-one approximate feasible solution. This relaxation approach is validated to be superior to the commercial solver Gurobi for \textbf{132} benchmark instances in terms of the relative gap between the generated objective value and the known best one and the number of instances with better objective values.

Error Bounds for Rank-one Double Nonnegative Reformulations of QAP and Exact Penalties

TL;DR

This work addresses the quadratic assignment problem by focusing on three equivalent rank-one doubly nonnegative reformulations. It establishes locally Lipschitz error bounds for the associated feasible set and proves global exact penalties for both the DC reformulation and BM factorizations, enabling a practical relaxation approach. An ALM-based relaxation method, EPalm, is developed to search for rank-one approximate feasible solutions, and numerical experiments on QAPLIB and Drezner instances show EPalm often delivers superior objective values compared with Gurobi within the same time budget. The results remove the need for calmness assumptions in prior penalty analyses and provide a scalable path toward high-quality feasible solutions for large QAP instances, with clear directions for future improvement in scalability and bound refinement.

Abstract

This paper focuses on the error bounds for several equivalent rank-one doubly nonnegative (DNN) conic reformulations of the quadratic assignment problem (QAP), a class of challenging combinatorial optimization problems. We provide three equivalent rank-one DNN reformulations of the QAP, including the one proposed in \cite{Jiang21}, and establish the locally and globally Lipschitzian error bounds for their feasible sets. Then, these error bounds are employed to prove that the penalty problems induced by the difference-of-convexity (DC) reformulation of the rank-one constraint are global exact penalties, and so are the penalty problems for their Burer-Monteiro (BM) factorizations. As a byproduct, the penalty problem for the rank-one DNN reformulation in \cite{Jiang21} is shown to be a global exact penalty without the calmness assumption. Finally, we illustrate the application of these exact penalties by proposing a relaxation approach with one of them to seek a rank-one approximate feasible solution. This relaxation approach is validated to be superior to the commercial solver Gurobi for \textbf{132} benchmark instances in terms of the relative gap between the generated objective value and the known best one and the number of instances with better objective values.
Paper Structure (15 sections, 12 theorems, 87 equations, 3 figures, 5 tables, 2 algorithms)

This paper contains 15 sections, 12 theorems, 87 equations, 3 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Main contributions
  4. Notation
  5. Preliminaries
  6. Calmness and subregularity
  7. Partial calmness of optimization problems
  8. Lipschitzian error bounds
  9. Global exact penalties
  10. Relaxation approach based on global exact penalty
  11. ALM for solving penalty subproblems
  12. Numerical experiment
  13. Numerical results for QAPLIB data
  14. Numerical results for Dre problems
  15. Conclusion

Key Result

Lemma 2.2

\newlabellemma-calm0 If $\vartheta$ is locally Lipschitz continuous relative to $\Delta$ and the mapping $\mathcal{S}$ in Definition def-pcalm0 is calm at $0$ for any $z\in\mathcal{Z}^*$, then $({\rm MP})$ is partially calm on $\mathcal{Z}^*$.

Figures (3)

  • Figure 1: Influence of $m$ on the objective value given by EPalm and its running time
  • Figure 2: Influence of $\rho_1$ and $\sigma$ on the objective value and rank returned by EPalm
  • Figure 3: The gap difference of Gurobi from that of EPalm for the QAPLIB instances (where Ggap appearing in ordinate is the one yielded by Gurobi)

Theorems & Definitions (28)

  • Definition 2.1
  • Lemma 2.2
  • Proof 1
  • Lemma 3.1
  • Proof 2
  • Proposition 3.2
  • Proof 3
  • Proposition 3.3
  • Proof 4
  • Proposition 3.4
  • ...and 18 more