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Bound reduction using pairs of linear inequalities

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  • Pietro Belotti

Abstract

We describe a procedure to reduce variable bounds in mixed integer nonlinear programming (MINLP) as well as mixed integer linear programming (MILP) problems. The procedure works by combining pairs of inequalities of a linear programming (LP) relaxation of the problem. This bound reduction procedure extends the feasibility based bound reduction technique on linear functions, used in MINLP and MILP. However, it can also be seen as a special case of optimality based bound reduction, a method to infer variable bounds from an LP relaxation of the problem. For an LP relaxation with m constraints and n variables, there are O(m 2 ) pairs of constraints, and a naïve implementation of our bound reduction scheme has complexity O(n 3 ) for each pair. Therefore, its overall complexity O(m 2 n 3 ) can be prohibitive for relatively large problems. We have developed a more efficient procedure that has complexity O(m 2 n 2 ), and embedded it in two Open-Source solvers: one for MINLP and one for MILP. We provide computational results which substantiate the usefulness of this bound reduction technique for several instances. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • Pietro Belotti, 2013. "Bound reduction using pairs of linear inequalities," Journal of Global Optimization, Springer, vol. 56(3), pages 787-819, July.
    • Handle: RePEc:spr:jglopt:v:56:y:2013:i:3:p:787-819
    DOI: 10.1007/s10898-012-9848-9
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    References listed on IDEAS

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    1. M. W. P. Savelsbergh, 1994. "Preprocessing and Probing Techniques for Mixed Integer Programming Problems," INFORMS Journal on Computing, INFORMS, vol. 6(4), pages 445-454, November.
    2. Kumar Abhishek & Sven Leyffer & Jeff Linderoth, 2010. "FilMINT: An Outer Approximation-Based Solver for Convex Mixed-Integer Nonlinear Programs," INFORMS Journal on Computing, INFORMS, vol. 22(4), pages 555-567, November.
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    1. Brais González-Rodríguez & Joaquín Ossorio-Castillo & Julio González-Díaz & Ángel M. González-Rueda & David R. Penas & Diego Rodríguez-Martínez, 2023. "Computational advances in polynomial optimization: RAPOSa, a freely available global solver," Journal of Global Optimization, Springer, vol. 85(3), pages 541-568, March.
    2. Ambros M. Gleixner & Timo Berthold & Benjamin Müller & Stefan Weltge, 2017. "Three enhancements for optimization-based bound tightening," Journal of Global Optimization, Springer, vol. 67(4), pages 731-757, April.
    3. Yifu Chen & Christos T. Maravelias, 2022. "Variable Bound Tightening and Valid Constraints for Multiperiod Blending," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2073-2090, July.
    4. Patrick Gemander & Wei-Kun Chen & Dieter Weninger & Leona Gottwald & Ambros Gleixner & Alexander Martin, 2020. "Two-row and two-column mixed-integer presolve using hashing-based pairing methods," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 8(3), pages 205-240, October.
    5. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.
    6. Harsha Nagarajan & Mowen Lu & Site Wang & Russell Bent & Kaarthik Sundar, 2019. "An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs," Journal of Global Optimization, Springer, vol. 74(4), pages 639-675, August.

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