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