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.
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Richard, JP.P., Tawarmalani, M. Lifting inequalities: a framework for generating strong cuts for nonlinear programs. Math. Program. 121, 61–104 (2010). https://doi.org/10.1007/s10107-008-0226-9
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DOI: https://doi.org/10.1007/s10107-008-0226-9
Keywords
- Nonlinear mixed-integer programming
- Cutting planes
- Bilinear knapsacks
- Convex extensions
- Sequence-independent lifting
- Elementary closures