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

Authors:
Jean-Philippe P. Richard;Mohit Tawarmalani
Affiliations:
Purdue University, School of Industrial Engineering, 315 N. Grant Street, 47907-2023, West Lafayette, IN, USA;Purdue University, Krannert School of Management, 100 S. Grant Street, 47907-2076, West Lafayette, IN, USA
Venue:
Mathematical Programming: Series A and B
Year:
2010

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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.