Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Centering ADMM for the Semidefinite Relaxation of the QAP

Shin-ichi Kanoh, Akiko Yoshise

TL;DR

The paper addresses solving the SDP relaxation of the quadratic assignment problem (QAP) by introducing Centering ADMM, a method that adds a barrier-based centering stage to the ADMM framework and then switches to standard ADMM when near the central path, all while holding the penalty parameter $\rho$ fixed. It proves global convergence of Centering ADMM under standard assumptions and a saddle-point condition for the unaugmented problem, leveraging a KKT-based argument to establish existence of a saddle point. Empirically, it shows that centering can substantially improve lower bounds on several QAPLIB classes (e.g., chr, Kra, Rou, Scr, Tai-a, Els19, Tho30) while being less advantageous for others (e.g., Had, Nug) in early iterations, highlighting a problem-dependent benefit. The results suggest Centering ADMM as a scalable alternative to interior-point methods for SDP relaxations of the QAP, with potential to accelerate bound computations in practice.

Abstract

We propose a new method for solving the semidefinite (SD) relaxation of the quadratic assignment problem (QAP), called Centering ADMM. Centering ADMM is an alternating direction method of multipliers (ADMM) combining the centering steps used in the interior-point method. The first stage of Centering ADMM updates the iterate so that it approaches the central path by incorporating a barrier function term into the objective function, as in the interior-point method. If the current iterate is sufficiently close to the central path with a sufficiently small value of the barrier parameter, the method switches to the standard version of ADMM. We show that Centering ADMM (not employing a dynamic update of the penalty parameter) has global convergence properties. To observe the effect of the centering steps, we conducted numerical experiments with SD relaxation problems of instances in QAPLIB. The results demonstrate that the centering steps are quite efficient for some classes of instances.

Centering ADMM for the Semidefinite Relaxation of the QAP

TL;DR

The paper addresses solving the SDP relaxation of the quadratic assignment problem (QAP) by introducing Centering ADMM, a method that adds a barrier-based centering stage to the ADMM framework and then switches to standard ADMM when near the central path, all while holding the penalty parameter fixed. It proves global convergence of Centering ADMM under standard assumptions and a saddle-point condition for the unaugmented problem, leveraging a KKT-based argument to establish existence of a saddle point. Empirically, it shows that centering can substantially improve lower bounds on several QAPLIB classes (e.g., chr, Kra, Rou, Scr, Tai-a, Els19, Tho30) while being less advantageous for others (e.g., Had, Nug) in early iterations, highlighting a problem-dependent benefit. The results suggest Centering ADMM as a scalable alternative to interior-point methods for SDP relaxations of the QAP, with potential to accelerate bound computations in practice.

Abstract

We propose a new method for solving the semidefinite (SD) relaxation of the quadratic assignment problem (QAP), called Centering ADMM. Centering ADMM is an alternating direction method of multipliers (ADMM) combining the centering steps used in the interior-point method. The first stage of Centering ADMM updates the iterate so that it approaches the central path by incorporating a barrier function term into the objective function, as in the interior-point method. If the current iterate is sufficiently close to the central path with a sufficiently small value of the barrier parameter, the method switches to the standard version of ADMM. We show that Centering ADMM (not employing a dynamic update of the penalty parameter) has global convergence properties. To observe the effect of the centering steps, we conducted numerical experiments with SD relaxation problems of instances in QAPLIB. The results demonstrate that the centering steps are quite efficient for some classes of instances.

Paper Structure

This paper contains 17 sections, 2 theorems, 50 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Standard ADMM
  3. Standard ADMM for the SD relaxation of the QAP
  4. Representation of the QAP for which the Standard ADMM is applicable
  5. Update formula of the variables in the Standard ADMM
  6. Stopping conditions of the Standard ADMM
  7. Centering ADMM
  8. Global convergence of Centering ADMM
  9. Numerical experiments
  10. Observations for the class of "chr" instances (Figure \ref{['fig:chr']})
  11. Observations for the class of "Had" instances (Figure \ref{['fig:Had']})
  12. Observations for the class of "Kra" instances (Figure \ref{['fig:Kra']})
  13. Observations for the classes of "Rou" and "Scr" instances (Figure \ref{['fig:Rou']})
  14. Observations for the class of "Nug" instances (Figure \ref{['fig:Nug']})
  15. Observations for the class of "Tai-a" instances (Figure \ref{['fig:Tai-a']})
  16. ...and 2 more sections

Key Result

Proposition 4.1

Suppose that we obtain a spectral decomposition of the matrix $\hat{V}^T Z \hat{V}\ + \rho\hat{V}^T Y \hat{V}$ into an orthogonal matrix $P$ and a diagonal matrix $D$, as Then, the new iterate $R^{k+1}$ in (eq:CADMM-R1) is given by where the matrix $\bar{R}$ is a diagonal matrix whose elements are

Figures (9)

  • Figure 1: Results for "chr" instances
  • Figure 16: Results for "Had" instances
  • Figure 22: Results for "Kra" instances
  • Figure 26: Results for "Rou" instances
  • Figure 30: Results for "Scr" instances
  • ...and 4 more figures

Theorems & Definitions (4)

  • Proposition 4.1
  • proof
  • Proposition 5.1
  • proof