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Affine FR: an Effective Facial Reduction Algorithm for Semidefinite Relaxations of Combinatorial Problems

Hao Hu, Boshi Yang

TL;DR

This work introduces affine FR, a fully automatic preprocessing method that stabilizes and accelerates SDP relaxations for combinatorial optimization problems with binary variables by restoring Slater's condition through a single affine-facet reduction. It leverages the affine hull of a problem-affine superset to construct a reduced facially reduced SDP via $Y=VRV^T$, diminishing matrix sizes from ${n+1}$ to ${r+1}$. Theoretical comparisons with analytical methods, partial FR, and Sieve-SDP show affine FR often matches or surpasses these approaches in reduction effectiveness, while remaining broadly applicable to CO relaxations such as Shor's and Lovász–Schrijver relaxations. Empirical results on 332 MIPLIB2017 mixed-binary instances demonstrate substantial reductions in matrix dimension and notable speedups, with improved numerical stability in cases where Slater’s condition would otherwise fail. Overall, affine FR enhances practicality and scalability of SDP-based CO solution methods by enabling more robust preprocessing with minimal user intervention.

Abstract

We develop a new method called affine facial reduction (FR) for recovering Slater's condition for semidefinite programming (SDP) relaxations of combinatorial optimization (CO) problems. Affine FR is a user-friendly method, as it is fully automatic and only requires a description of the problem. We provide a rigorous analysis of differences between affine FR and the existing methods. We also present numerical results to demonstrate the effectiveness of affine FR in reducing the size of SDP relaxations for CO problems.

Affine FR: an Effective Facial Reduction Algorithm for Semidefinite Relaxations of Combinatorial Problems

TL;DR

This work introduces affine FR, a fully automatic preprocessing method that stabilizes and accelerates SDP relaxations for combinatorial optimization problems with binary variables by restoring Slater's condition through a single affine-facet reduction. It leverages the affine hull of a problem-affine superset to construct a reduced facially reduced SDP via , diminishing matrix sizes from to . Theoretical comparisons with analytical methods, partial FR, and Sieve-SDP show affine FR often matches or surpasses these approaches in reduction effectiveness, while remaining broadly applicable to CO relaxations such as Shor's and Lovász–Schrijver relaxations. Empirical results on 332 MIPLIB2017 mixed-binary instances demonstrate substantial reductions in matrix dimension and notable speedups, with improved numerical stability in cases where Slater’s condition would otherwise fail. Overall, affine FR enhances practicality and scalability of SDP-based CO solution methods by enabling more robust preprocessing with minimal user intervention.

Abstract

We develop a new method called affine facial reduction (FR) for recovering Slater's condition for semidefinite programming (SDP) relaxations of combinatorial optimization (CO) problems. Affine FR is a user-friendly method, as it is fully automatic and only requires a description of the problem. We provide a rigorous analysis of differences between affine FR and the existing methods. We also present numerical results to demonstrate the effectiveness of affine FR in reducing the size of SDP relaxations for CO problems.
Paper Structure (23 sections, 13 theorems, 76 equations, 4 figures, 9 tables)

This paper contains 23 sections, 13 theorems, 76 equations, 4 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Notation.
  3. Preliminaries
  4. SDP relaxation for CO problems
  5. Shor's SDP relaxation
  6. Facial reduction
  7. Polyhedral theory and linear algebra
  8. Explicit expression for $SEP(L \cap \mathcal{K})$
  9. Affine FR
  10. Computing the affine hull
  11. Application to the Lovász-Schrijver relaxation
  12. A Theoretical Comparison
  13. Analytical methods
  14. Partial FR
  15. Non-negative diagonal matrices
  16. ...and 8 more sections

Key Result

Lemma 1

An inequality $c^{T}x \leq \delta$ is valid for $P$ if and only if there exists $u \geq \textbf{0}$ such that $u^{T}A = c^{T}$ and $u^{T}b \leq \delta$.

Figures (4)

  • Figure 1: The red dots constitute $F$, and the grey triangular area represents $P$.
  • Figure 2: The affine hull of $P$ is represented by the (unbounded) gray area.
  • Figure 3: The red dots constitute $F$.
  • Figure 4: The affine hull of $F$ is represented by the line connecting the two red points.

Theorems & Definitions (30)

  • Lemma 1: conforti2014integer
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Example 1
  • Example 2
  • remark thmcounterremark
  • Theorem 1
  • Lemma 4
  • ...and 20 more