Mathematical Programming

, Volume 121, Issue 1, pp 61–104 | Cite as

Lifting inequalities: a framework for generating strong cuts for nonlinear programs

FULL LENGTH PAPER Series A

Abstract

In this paper, we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs. The theory is geometric and provides insights into lifting-based cut generation procedures, yielding short proofs of earlier results in mixed-integer programming. Using convex extensions, we obtain conditions that allow for sequence-independent lifting in nonlinear settings, paving a way for efficient cut-generation procedures for nonlinear programs. This sequence-independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general-purpose, strong cuts for integer programs. We specialize our lifting results to derive facet-defining inequalities for mixed-integer bilinear knapsack sets. Finally, we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut-generation techniques applied on a tight reformulation of the problem.

Keywords

Nonlinear mixed-integer programming Cutting planes Bilinear knapsacks Convex extensions Sequence-independent lifting Elementary closures 

Mathematics Subject Classification (2000)

90C26 90C30 90C11 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.School of Industrial EngineeringPurdue UniversityWest LafayetteUSA
  2. 2.Krannert School of ManagementPurdue UniversityWest LafayetteUSA

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