IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v84y2022i4d10.1007_s10898-022-01184-6.html
   My bibliography  Save this article

Compact mixed-integer programming formulations in quadratic optimization

Author

Listed:
  • Benjamin Beach

    (Virginia Tech)

  • Robert Hildebrand

    (Virginia Tech)

  • Joey Huchette

    (Rice University)

Abstract

We present a technique for producing valid dual bounds for nonconvex quadratic optimization problems. The approach leverages an elegant piecewise linear approximation for univariate quadratic functions due to Yarotsky (Neural Netw 94:103–114, 2017), formulating this (simple) approximation using mixed-integer programming (MIP). Notably, the number of constraints, binary variables, and auxiliary continuous variables used in this formulation grows logarithmically in the approximation error. Combining this with a diagonal perturbation technique to convert a nonseparable quadratic function into a separable one, we present a mixed-integer convex quadratic relaxation for nonconvex quadratic optimization problems. We study the strength (or sharpness) of our formulation and the tightness of its approximation. Further, we show that our formulation represents feasible points via a Gray code. We close with computational results on problems with quadratic objectives and/or constraints, showing that our proposed method (i) across the board outperforms existing MIP relaxations from the literature, and (ii) on hard instances produces better bounds than exact solvers within a fixed time budget.

Suggested Citation

  • Benjamin Beach & Robert Hildebrand & Joey Huchette, 2022. "Compact mixed-integer programming formulations in quadratic optimization," Journal of Global Optimization, Springer, vol. 84(4), pages 869-912, December.
  • Handle: RePEc:spr:jglopt:v:84:y:2022:i:4:d:10.1007_s10898-022-01184-6
    DOI: 10.1007/s10898-022-01184-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-022-01184-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10898-022-01184-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Pedro A. Castillo Castillo & Pedro M. Castro & Vladimir Mahalec, 2018. "Global optimization of MIQCPs with dynamic piecewise relaxations," Journal of Global Optimization, Springer, vol. 71(4), pages 691-716, August.
    2. Juan Pablo Vielma & Shabbir Ahmed & George Nemhauser, 2010. "Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions," Operations Research, INFORMS, vol. 58(2), pages 303-315, April.
    3. Santanu S. Dey & Akshay Gupte, 2015. "Analysis of MILP Techniques for the Pooling Problem," Operations Research, INFORMS, vol. 63(2), pages 412-427, April.
    4. Fred Glover, 1975. "Improved Linear Integer Programming Formulations of Nonlinear Integer Problems," Management Science, INFORMS, vol. 22(4), pages 455-460, December.
    5. Pierre Hansen & Brigitte Jaumard & MichèLe Ruiz & Junjie Xiong, 1993. "Global minimization of indefinite quadratic functions subject to box constraints," Naval Research Logistics (NRL), John Wiley & Sons, vol. 40(3), pages 373-392, April.
    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.
    7. LEE, Jon & WILSON, Dan, 2001. "Polyhedral methods for piecewise-linear functions I: the lambda method," LIDAM Reprints CORE 1493, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jon Lee & Daphne Skipper & Emily Speakman & Luze Xu, 2023. "Gaining or Losing Perspective for Piecewise-Linear Under-Estimators of Convex Univariate Functions," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 1-35, January.
    2. Moritz Link & Stefan Volkwein, 2023. "Adaptive piecewise linear relaxations for enclosure computations for nonconvex multiobjective mixed-integer quadratically constrained programs," Journal of Global Optimization, Springer, vol. 87(1), pages 97-132, September.
    3. Alexander J. Zolan & Michael S. Scioletti & David P. Morton & Alexandra M. Newman, 2021. "Decomposing Loosely Coupled Mixed-Integer Programs for Optimal Microgrid Design," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1300-1319, October.
    4. Silva, Thiago Lima & Camponogara, Eduardo, 2014. "A computational analysis of multidimensional piecewise-linear models with applications to oil production optimization," European Journal of Operational Research, Elsevier, vol. 232(3), pages 630-642.
    5. Srikrishna Sridhar & Jeffrey Linderoth & James Luedtke, 2014. "Models and solution techniques for production planning problems with increasing byproducts," Journal of Global Optimization, Springer, vol. 59(2), pages 597-631, July.
    6. Juan Pablo Vielma, 2018. "Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139," Management Science, INFORMS, vol. 64(10), pages 4721-4734, October.
    7. Birolini, Sebastian & Jacquillat, Alexandre & Cattaneo, Mattia & Antunes, António Pais, 2021. "Airline Network Planning: Mixed-integer non-convex optimization with demand–supply interactions," Transportation Research Part B: Methodological, Elsevier, vol. 154(C), pages 100-124.
    8. Natashia Boland & Thomas Kalinowski & Fabian Rigterink, 2016. "New multi-commodity flow formulations for the pooling problem," Journal of Global Optimization, Springer, vol. 66(4), pages 669-710, December.
    9. Codas, Andrés & Camponogara, Eduardo, 2012. "Mixed-integer linear optimization for optimal lift-gas allocation with well-separator routing," European Journal of Operational Research, Elsevier, vol. 217(1), pages 222-231.
    10. Christensen, Tue R.L. & Labbé, Martine, 2015. "A branch-cut-and-price algorithm for the piecewise linear transportation problem," European Journal of Operational Research, Elsevier, vol. 245(3), pages 645-655.
    11. Yokoyama, Ryohei & Kitano, Hiroyuki & Wakui, Tetsuya, 2017. "Optimal operation of heat supply systems with piping network," Energy, Elsevier, vol. 137(C), pages 888-897.
    12. Tian, Xueyu & You, Fengqi, 2019. "Carbon-neutral hybrid energy systems with deep water source cooling, biomass heating, and geothermal heat and power," Applied Energy, Elsevier, vol. 250(C), pages 413-432.
    13. Longinidis, Pantelis & Georgiadis, Michael C., 2014. "Integration of sale and leaseback in the optimal design of supply chain networks," Omega, Elsevier, vol. 47(C), pages 73-89.
    14. Rostami, Borzou & Chassein, André & Hopf, Michael & Frey, Davide & Buchheim, Christoph & Malucelli, Federico & Goerigk, Marc, 2018. "The quadratic shortest path problem: complexity, approximability, and solution methods," European Journal of Operational Research, Elsevier, vol. 268(2), pages 473-485.
    15. Unai Aldasoro & María Merino & Gloria Pérez, 2019. "Time consistent expected mean-variance in multistage stochastic quadratic optimization: a model and a matheuristic," Annals of Operations Research, Springer, vol. 280(1), pages 151-187, September.
    16. Christodoulos Floudas & Xiaoxia Lin, 2005. "Mixed Integer Linear Programming in Process Scheduling: Modeling, Algorithms, and Applications," Annals of Operations Research, Springer, vol. 139(1), pages 131-162, October.
    17. Gupta, Renu & Bandopadhyaya, Lakshmisree & Puri, M. C., 1996. "Ranking in quadratic integer programming problems," European Journal of Operational Research, Elsevier, vol. 95(1), pages 231-236, November.
    18. Angel L. Cedeño & Reinier López Ahuar & José Rojas & Gonzalo Carvajal & César Silva & Juan C. Agüero, 2022. "Model Predictive Control for Photovoltaic Plants with Non-Ideal Energy Storage Using Mixed Integer Linear Programming," Energies, MDPI, vol. 15(17), pages 1-21, September.
    19. Osman, Hany & Demirli, Kudret, 2010. "A bilinear goal programming model and a modified Benders decomposition algorithm for supply chain reconfiguration and supplier selection," International Journal of Production Economics, Elsevier, vol. 124(1), pages 97-105, March.
    20. Verbiest, Floor & Cornelissens, Trijntje & Springael, Johan, 2019. "A matheuristic approach for the design of multiproduct batch plants with parallel production lines," European Journal of Operational Research, Elsevier, vol. 273(3), pages 933-947.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:84:y:2022:i:4:d:10.1007_s10898-022-01184-6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.