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The dual-path fixing strategy and its application to the set-covering problem

Paulo Michel F. Yamagishi, Marcia Fampa, Jon Lee

TL;DR

This work introduces dual-path fixing (DPF), a strategy that leverages the sequence of dual-feasible solutions generated during relaxations to fix more binary variables within branch-and-bound than traditional single-solution approaches, while keeping overhead near that of solving the relaxation. Building on Lagrangian duality and relations to reduced-cost fixing and strong fixing, DPF aggregates dual information to derive fixing decisions for 0/1 MILO and MINLO problems, with specific focus on set-covering MILO. The authors derive principled fixing rules, illustrate the approach for pure 0/1 MILO, and demonstrate, via numerical experiments on Beasley SCP and eVTOL SLS instances, that DPF increases the number of fixed variables and approaches the performance of strong fixing at substantially lower cost. They also discuss preprocessing via dominated-row elimination and emphasize the need for tighter integration of subsolvers with B&B in modern MILO solvers to realize DPF’s practical impact, with several promising directions for extension to MINLO and conic relaxations. Overall, the work advances exact algorithmic tooling for MILO/MINO by combining dual information across iterative relaxations to accelerate branching decisions and reduce problem size.

Abstract

We introduce the dual-path fixing strategy to exploit dual algorithms for solving relaxations of mixed-integer nonlinear-optimization problems. Such dual algorithms are naturally applied in the context of branch-and-bound, and eventual impact on the success of branch-and-bound is our strong motivation. Our fixing strategy aims to be more powerful than the common strategy of fixing variables based on a single dual-feasible solution (e.g., standard reduced-cost fixing for mixed-integer linear optimization), but to be much faster than ``strong fixing'', essentially requiring no more time than that of the dual algorithm that we exploit. We have successfully tested our ideas on mixed-integer linear-optimization set-covering instances from the literature, in the context of the dual-simplex method applied to the continuous relaxations.

The dual-path fixing strategy and its application to the set-covering problem

TL;DR

This work introduces dual-path fixing (DPF), a strategy that leverages the sequence of dual-feasible solutions generated during relaxations to fix more binary variables within branch-and-bound than traditional single-solution approaches, while keeping overhead near that of solving the relaxation. Building on Lagrangian duality and relations to reduced-cost fixing and strong fixing, DPF aggregates dual information to derive fixing decisions for 0/1 MILO and MINLO problems, with specific focus on set-covering MILO. The authors derive principled fixing rules, illustrate the approach for pure 0/1 MILO, and demonstrate, via numerical experiments on Beasley SCP and eVTOL SLS instances, that DPF increases the number of fixed variables and approaches the performance of strong fixing at substantially lower cost. They also discuss preprocessing via dominated-row elimination and emphasize the need for tighter integration of subsolvers with B&B in modern MILO solvers to realize DPF’s practical impact, with several promising directions for extension to MINLO and conic relaxations. Overall, the work advances exact algorithmic tooling for MILO/MINO by combining dual information across iterative relaxations to accelerate branching decisions and reduce problem size.

Abstract

We introduce the dual-path fixing strategy to exploit dual algorithms for solving relaxations of mixed-integer nonlinear-optimization problems. Such dual algorithms are naturally applied in the context of branch-and-bound, and eventual impact on the success of branch-and-bound is our strong motivation. Our fixing strategy aims to be more powerful than the common strategy of fixing variables based on a single dual-feasible solution (e.g., standard reduced-cost fixing for mixed-integer linear optimization), but to be much faster than ``strong fixing'', essentially requiring no more time than that of the dual algorithm that we exploit. We have successfully tested our ideas on mixed-integer linear-optimization set-covering instances from the literature, in the context of the dual-simplex method applied to the continuous relaxations.
Paper Structure (17 sections, 2 theorems, 17 equations, 9 figures, 2 tables)

This paper contains 17 sections, 2 theorems, 17 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Fixing variables in binary nonlinear programs
  3. Lagrangian duality and fixing for 0/1 MINLO
  4. Strong fixing for 0/1 MINLO
  5. Dual-path fixing for 0/1 MINLO
  6. Particularization to pure-0/1 MILO
  7. Strong fixing for pure-0/1 MILO
  8. Dual-path fixing for 0/1 MILO
  9. Set-covering MILO
  10. Variable fixing for set covering
  11. Dominated row elimination (DRE) for set covering
  12. Numerical experiments for set covering
  13. Procedures evaluated
  14. Test instances
  15. Results
  16. ...and 2 more sections

Key Result

Theorem 1

Let Then, for every optimal solution $(x^*,y^*)$ for BNLP, we have:

Figures (9)

  • Figure 1: Average percentage reduction in the number of variables (Beasley instances - sets 4--6)
  • Figure 2: Average percentage reduction in the number of variables (Beasley instances - sets A--D)
  • Figure 3: Average percentage reduction in the number of variables (SLS instances)
  • Figure 4: DPF behavior per simplex iteration - SLS instance originally with $n=1000, m=7585$
  • Figure 5: DPF behavior per simplex iteration - Beasley instances - sets 4--6
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Remark 2
  • Theorem 3