Lift-and-project cuts for convex mixed integer nonlinear programs Linear programming based separation and extended formulations

We describe a computationally effective method for generating lift-and-project cuts for convex mixed-integer nonlinear programs (MINLPs). The method relies on solving a sequence of cut-generating linear programs and in the limit generates an inequality as strong as the lift-and-project cut that can...

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Bibliographic Details
Published in:Mathematical programming computation Vol. 9; no. 4; pp. 499 - 526
Main Authors: Kılınç, Mustafa R., Linderoth, Jeff, Luedtke, James
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2017
Subjects:
Full Length Paper
Mathematics
Mathematics and Statistics
Mathematics of Computing
Operations Research/Decision Theory
Optimization
Theory of Computation
Extended formulations
90C30
Mixed-integer nonlinear programming
90C57
Lift-and-project cuts
90C11
ISSN:1867-2949, 1867-2957
Online Access:Get full text
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Summary:We describe a computationally effective method for generating lift-and-project cuts for convex mixed-integer nonlinear programs (MINLPs). The method relies on solving a sequence of cut-generating linear programs and in the limit generates an inequality as strong as the lift-and-project cut that can be obtained from solving a cut-generating nonlinear program. Using this procedure, we are able to approximately optimize over the rank one lift-and-project closure for a variety of convex MINLP instances. The results indicate that lift-and-project cuts have the potential to close a significant portion of the integrality gap for convex MINLPs. In addition, we find that using this procedure within a branch-and-cut solver for convex MINLPs significantly reduces the total solution time for many instances. We also demonstrate that combining lift-and-project cuts with an extended formulation that exploits separability of convex functions yields significant improvements in both relaxation bounds and the time to calculate the relaxation. Overall, these results suggest that with an effective separation routine, like the one proposed here, lift-and-project cuts may be as effective for solving convex MINLPs as they have been for solving mixed-integer linear programs.
ISSN:1867-2949
1867-2957
DOI:10.1007/s12532-017-0118-1