Solving Lift-and-Project Relaxations of Binary Integer Programs

Abstract

We propose a method for optimizing the lift-and-project relaxations of binary integer programs introduced by Lovász and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a restructuring of the relaxations, which isolates the complicating constraints and allows for a Lagrangian approach. We detail an enhanced subgradient method and discuss its efficient implementation. Computational results illustrate that our algorithm produces tight bounds more quickly than state-of-the-art linear and semidefinite solvers. Introduction.In the field of optimization, binary integer programs have proven to be an excellent source of challenging problems, and the successful solution of larger and larger problems over the past few decades has required significant theoretical and computational advances. One of the fundamental issues is how to obtain a "good" description of the convex hull of integer solutions, and many specific classes of integer programs have been solved by finding problem-specific ways to address this issue.Researchers have also developed techniques for approximating the convex hull of integer solutions without any specific knowledge of the problem, i.e., techniques that apply to arbitrary binary integer programs. Some of the earliest work done in this direction was by Gomory [19] in generating linear inequalities that tighten the basic linear relaxation. A different idea, which has been advocated by several authors, is to approximate the convex hull as the projection of some polyhedron lying in a space of higher dimension. We refer the reader to [3,38,32,4,24,7]. Connections between these works are explored in [27,26].Although these so-called lift-and-project methods are quite powerful theoretically, they present great computational challenges because one typically must optimize in the space of the lifting, i.e., the space of higher dimension. Computational issues are detailed in [4,39,11,22,14].In this paper, we focus on the techniques proposed by Lovász and Schrijver (LS), including both linear and semidefinite relaxations. In particular, our main goal is to present improved computational methods for optimizing over the first-level LS relaxations. We are aware of only one study (by Dash [14]), which investigates the strength of these relaxations computationally. This shortage of computational experience is due to the dramatic size of these relaxations. For example, one specific semidefinite

Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting

“…A popular way to derive the SDRs of QAP is to relax the rank-1 matrix vec(X)vec(X) T to a n 2 × n 2 positive semidefinite matrix with nonnegative elements, where vec(X) denotes the n 2 -dimensional vector obtained from X by stacking its columns sequentially into a long vector. Though much progress has been obtained in solving the SDR based on the gram matrix vec(X)vec(X) T [29,5,10], the large number of O(n 4 ) variables and constraints in these relaxations still make them formidable for medium size QAP instances with the current computation facilities. Recently, Ding and Wolkowicz [11] introduced a new SDR of QAP based on matrix lifting.…”

Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem

“…It is also the same as the so-called N + (K )-relaxation of Lovász and Schrijver [17] applied to the QAP, as studied by Burer and Vandenbussche [4]. The equivalence between the two relaxations was recently shown by Povh and Rendl [19].…”

“…In [4] the optimal value of the SDP relaxation (13) was approximately computed for several instances of the QAPLIB library using an augmented Lagragian method. These values, rounded up, are given in the column 'previous l.b.'…”

“…This has led to the application of bundle methods [20] and augmented Lagrangian methods [4] to certain SDP relaxations of the QAP. Concerning one SDP relaxation (that we will consider in this paper), the authors of [20] write that '... the relaxation ... cannot be solved straightforward [ly] by interior point methods for interesting instances (n ≥ 15).…”

A Level-3 Reformulation-Linearization Technique-Based Bound for the Quadratic Assignment Problem

“…Recent developments have produced improved, that is, tighter, bounds. The new methodologies include the interior point bound by Resende et al (1995), the level-1 RLT-based dual-ascent bound by Hahn and Grant (1998), the dual-based bound by Karisch et al (1999), the convex quadratic programming bound by Anstreicher and Brixius (2001), the level-2 RLT interior point bound by Ramakrishnan et al (2002), the SDP bound by Roupin (2004), the lift-and-project SDP bound by Burer and Vandenbussche (2006), the bundle method bound by Rendl and Sotirov (2007), and the HahnHightower level-2 RLT-based dual-ascent bound by Adams et al (2007). The tightest bounds are the lift-and-project SDP bound and the two level-2 RLT-based bounds.…”