IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v80y2021i2d10.1007_s10898-020-00979-9.html
   My bibliography  Save this article

Convexification techniques for linear complementarity constraints

Author

Listed:
  • Trang T. Nguyen

    (University of Florida)

  • Jean-Philippe P. Richard

    (University of Minnesota)

  • Mohit Tawarmalani

    (Purdue University)

Abstract

We develop convexification techniques for mathematical programs with complementarity constraints. Specifically, we adapt the reformulation-linearization technique of Sherali and Adams (SIAM J Discrete Math 3, 411–430, 1990) to problems with linear complementarity constraints and discuss how this procedure reduces to its standard specification for binary mixed-integer programs. Then, we consider specially structured complementarity sets that appear in KKT systems with linear constraints. For sets with a single complementarity constraint, we develop a convexification procedure that generates all nontrivial facet-defining inequalities and has an appealing “cancel-and-relax” interpretation. This procedure is used to describe the convex hull of problems with few side constraints in closed-form. As a consequence, we delineate cases when the factorable relaxation techniques yield the convex hull from those for which they do not. We then discuss how these results extend to sets with multiple complementarity constraints. In particular, we show that a suitable sequential application of the cancel-and-relax procedure produces all nontrivial inequalities of their convex hull. We conclude by illustrating, on a set of randomly generated problems, that the relaxations we propose can be significantly stronger than those available in the literature.

Suggested Citation

  • Trang T. Nguyen & Jean-Philippe P. Richard & Mohit Tawarmalani, 2021. "Convexification techniques for linear complementarity constraints," Journal of Global Optimization, Springer, vol. 80(2), pages 249-286, June.
  • Handle: RePEc:spr:jglopt:v:80:y:2021:i:2:d:10.1007_s10898-020-00979-9
    DOI: 10.1007/s10898-020-00979-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-020-00979-9
    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/s10898-020-00979-9?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. Toshihide Ibaraki, 1973. "Technical Note—The Use of Cuts in Complementary Programming," Operations Research, INFORMS, vol. 21(1), pages 353-359, February.
    2. Egon Balas, 1971. "Intersection Cuts—A New Type of Cutting Planes for Integer Programming," Operations Research, INFORMS, vol. 19(1), pages 19-39, February.
    3. William Cook & Sanjeeb Dash, 2001. "On the Matrix-Cut Rank of Polyhedra," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 19-30, February.
    4. Monique Laurent, 2003. "A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0--1 Programming," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 470-496, August.
    5. B. Ramarao & C. M. Shetty, 1984. "Application of disjunctive programming to the linear complementarity problem," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 31(4), pages 589-600, December.
    6. Jing Hu & John Mitchell & Jong-Shi Pang & Bin Yu, 2012. "On linear programs with linear complementarity constraints," Journal of Global Optimization, Springer, vol. 53(1), pages 29-51, May.
    7. I. R. De Farias & E. L. Johnson & G. L. Nemhauser, 2002. "Facets of the Complementarity Knapsack Polytope," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 210-226, February.
    8. H. D. Sherali & R. S. Krishnamurthy & F. A. Al-Khayyal, 1998. "Enumeration Approach for Linear Complementarity Problems Based on a Reformulation-Linearization Technique," Journal of Optimization Theory and Applications, Springer, vol. 99(2), pages 481-507, November.
    9. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Liang He & Xiaoqing Li & Shaohua Lei & Bo Bi & Suozhong Chen, 2023. "A Front Advancing Adaptive Triangular Mesh Dynamic Generation Algorithm and Its Application in 3D Geological Modeling," Sustainability, MDPI, vol. 15(9), pages 1-16, April.

    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. Francisco Jara-Moroni & John E. Mitchell & Jong-Shi Pang & Andreas Wächter, 2020. "An enhanced logical benders approach for linear programs with complementarity constraints," Journal of Global Optimization, Springer, vol. 77(4), pages 687-714, August.
    2. Pratik Worah, 2015. "Rank bounds for a hierarchy of Lovász and Schrijver," Journal of Combinatorial Optimization, Springer, vol. 30(3), pages 689-709, October.
    3. Gábor Braun & Samuel Fiorini & Sebastian Pokutta & David Steurer, 2015. "Approximation Limits of Linear Programs (Beyond Hierarchies)," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 756-772, March.
    4. Amitabh Basu & Robert Hildebrand & Matthias Köppe, 2016. "Light on the infinite group relaxation I: foundations and taxonomy," 4OR, Springer, vol. 14(1), pages 1-40, March.
    5. Amitabh Basu & Robert Hildebrand & Matthias Köppe, 2016. "Light on the infinite group relaxation II: sufficient conditions for extremality, sequences, and algorithms," 4OR, Springer, vol. 14(2), pages 107-131, June.
    6. Campi, Luciano & Zabaljauregui, Diego, 2020. "Optimal market making under partial information with general intensities," LSE Research Online Documents on Economics 104612, London School of Economics and Political Science, LSE Library.
    7. He, Wei & Sun, Yeneng, 2013. "Stationary Markov Perfect Equilibria in Discounted Stochastic Games," MPRA Paper 51274, University Library of Munich, Germany.
    8. Karan N. Chadha & Ankur A. Kulkarni, 2022. "On independent cliques and linear complementarity problems," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(4), pages 1036-1057, December.
    9. Eduardo Perez & Delphine Prady, 2012. "Complicating to Persuade?," Working Papers hal-03583827, HAL.
    10. Romain Blanchard & Laurence Carassus & Miklós Rásonyi, 2018. "No-arbitrage and optimal investment with possibly non-concave utilities: a measure theoretical approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(2), pages 241-281, October.
    11. René Aïd & Matteo Basei & Giorgia Callegaro & Luciano Campi & Tiziano Vargiolu, 2020. "Nonzero-Sum Stochastic Differential Games with Impulse Controls: A Verification Theorem with Applications," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 205-232, February.
    12. He, Wei & Yannelis, Nicholas C., 2015. "Equilibrium theory under ambiguity," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 86-95.
    13. Massimiliano Amarante & Mario Ghossoub & Edmund Phelps, 2012. "Contracting for Innovation under Knightian Uncertainty," Cahiers de recherche 18-2012, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    14. Sudhir A. Shah, 2016. "The Generalized Arrow-Pratt Coefficient," Working Papers id:10795, eSocialSciences.
    15. Luçon, Eric, 2020. "Quenched asymptotics for interacting diffusions on inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6783-6842.
    16. Lashi Bandara & Paul Bryan, 2020. "Heat kernels and regularity for rough metrics on smooth manifolds," Mathematische Nachrichten, Wiley Blackwell, vol. 293(12), pages 2255-2270, December.
    17. Carlos Pimienta & Jianfei Shen, 2014. "On the equivalence between (quasi-)perfect and sequential equilibria," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(2), pages 395-402, May.
    18. Oriol Carbonell-Nicolau, 2021. "Equilibria in infinite games of incomplete information," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(2), pages 311-360, June.
    19. Amarante, Massimiliano & Ghossoub, Mario & Phelps, Edmund, 2015. "Ambiguity on the insurer’s side: The demand for insurance," Journal of Mathematical Economics, Elsevier, vol. 58(C), pages 61-78.
    20. Toraubally, Waseem A., 2018. "Large market games, the law of one price, and market structure," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 13-26.

    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:80:y:2021:i:2:d:10.1007_s10898-020-00979-9. 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.