Mathematical Programming

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

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



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.


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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