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Lift-and-project cuts for convex mixed integer nonlinear programs

Linear programming based separation and extended formulations

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Abstract

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.

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Acknowledgements

This work was supported in part by the U.S. National Science Foundation (CCF-0830153) and by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contract numbers DE-AC02-06CH11357 and DE-FG02-08ER25861. The authors would like to thank Andrew Miller for his insightful comments on this work.

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Correspondence to Mustafa R. Kılınç.

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Appendix

Table 6 presents the gaps closed when using monoidal strenghthening to obtain strengthened lift-and-project cuts in the closure experiment presented in Sect. 6.3. The structure of this table is identical to that of Table 3. Table 3 in the Electronic Supplementary Material provides the results for each individual instance.

Table 6 Strengthened lift-and-project closure results summarized by instance family

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Kılınç, M.R., Linderoth, J. & Luedtke, J. Lift-and-project cuts for convex mixed integer nonlinear programs. Math. Prog. Comp. 9, 499–526 (2017). https://doi.org/10.1007/s12532-017-0118-1

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