IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v274y2019i1d10.1007_s10479-018-2926-8.html
   My bibliography  Save this article

Total dual integrality of the linear complementarity problem

Author

Listed:
  • Hanna Sumita

    (Tokyo Metropolitan University)

  • Naonori Kakimura

    (Keio University)

  • Kazuhisa Makino

    (Kyoto University)

Abstract

In this paper, we introduce total dual integrality of the linear complementarity problem (LCP) by analogy with the linear programming problem. The main idea of defining the notion is to propose the LCP with orientation, a variant of the LCP whose feasible complementary cones are specified by an additional input vector. Then we naturally define the dual problem of the LCP with orientation and total dual integrality of the LCP. We show that if the LCP is totally dual integral, then all basic solutions are integral. If the input matrix is sufficient or rank-symmetric, and the LCP is totally dual integral, then our result implies that the LCP always has an integral solution whenever it has a solution. We also introduce a class of matrices such that any LCP instance having the matrix as a coefficient matrix is totally dual integral. We investigate relationships between matrix classes in the LCP literature such as principally unimodular matrices. Principally unimodular matrices are known that all basic solutions to the LCP are integral for any integral input vector. In addition, we show that it is coNP-hard to decide whether a given LCP instance is totally dual integral.

Suggested Citation

  • Hanna Sumita & Naonori Kakimura & Kazuhisa Makino, 2019. "Total dual integrality of the linear complementarity problem," Annals of Operations Research, Springer, vol. 274(1), pages 531-553, March.
    • Handle: RePEc:spr:annopr:v:274:y:2019:i:1:d:10.1007_s10479-018-2926-8
    DOI: 10.1007/s10479-018-2926-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-018-2926-8
    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/s10479-018-2926-8?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. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(5), pages 687-698, October.
    2. C. E. Lemke, 1965. "Bimatrix Equilibrium Points and Mathematical Programming," Management Science, INFORMS, vol. 11(7), pages 681-689, May.
    3. R. Chandrasekaran & S. N. Kabadi & R. Sridhar, 1998. "Integer Solution for Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 390-402, May.
    4. William H. Cunningham & James F. Geelen, 1998. "Integral Solutions of Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 23(1), pages 61-68, February.
    5. EDMONDS, Jack & GILES, Rick, 1977. "A min-max relation for submodular functions on graphs," LIDAM Reprints CORE 301, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Joseph T. Howson, Jr., 1972. "Equilibria of Polymatrix Games," Management Science, INFORMS, vol. 18(5-Part-1), pages 312-318, January.
    7. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(3), pages 381-386, June.
    8. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(4), pages 525-537, August.
    9. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(2), pages 285-292, April.
    10. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
    11. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(1), pages 151-159, February.
    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. Hanna Sumita & Naonori Kakimura & Kazuhisa Makino, 2015. "The Linear Complementarity Problems with a Few Variables per Constraint," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 1015-1026, October.
    2. Yun-Bin Zhao & Duan Li, 2001. "On a New Homotopy Continuation Trajectory for Nonlinear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 119-146, February.
    3. Dolf Talman & Zaifu Yang, 2012. "On a Parameterized System of Nonlinear Equations with Economic Applications," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 644-671, August.
    4. Zhiqiang Zheng & Balaji Padmanabhan & Steven O. Kimbrough, 2003. "On the Existence and Significance of Data Preprocessing Biases in Web-Usage Mining," INFORMS Journal on Computing, INFORMS, vol. 15(2), pages 148-170, May.
    5. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999. "Variational Inequality Problems With a Continuum of Solutions : Existence and Computation," Other publications TiSEM 73e2f01b-ad4d-4447-95ba-a, Tilburg University, School of Economics and Management.
    6. Carlos R. Handy & Daniel Vrinceanu & Carl B. Marth & Harold A. Brooks, 2015. "Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure," Mathematics, MDPI, vol. 3(4), pages 1-24, November.
    7. Allen C. Goodman & Miron Stano, 2000. "Hmos and Health Externalities: A Local Public Good Perspective," Public Finance Review, , vol. 28(3), pages 247-269, May.
    8. Bode, Sven & Michaelowa, Axel, 2003. "Avoiding perverse effects of baseline and investment additionality determination in the case of renewable energy projects," Energy Policy, Elsevier, vol. 31(6), pages 505-517, May.
    9. Ala, Guido & Fasshauer, Gregory E. & Francomano, Elisa & Ganci, Salvatore & McCourt, Michael J., 2017. "An augmented MFS approach for brain activity reconstruction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 3-15.
    10. Bettina Campedelli & Andrea Guerrina & Giulia Romano & Chiara Leardini, 2014. "La performance della rete ospedaliera pubblica della regione Veneto. L?impatto delle variabili ambientali e operative sull?efficienza," MECOSAN, FrancoAngeli Editore, vol. 2014(92), pages 119-142.
    11. Haider A. Khan, 2004. "General Conclusions: From Crisis to a Global Political Economy of Freedom," Palgrave Macmillan Books, in: Global Markets and Financial Crises in Asia, chapter 9, pages 193-211, Palgrave Macmillan.
    12. Penn Loh & Zoë Ackerman & Joceline Fidalgo & Rebecca Tumposky, 2022. "Co-Education/Co-Research Partnership: A Critical Approach to Co-Learning between Dudley Street Neighborhood Initiative and Tufts University," Social Sciences, MDPI, vol. 11(2), pages 1-17, February.
    13. Broekhuis, Manda & Vos, Janita F.J., 2003. "Improving organizational sustainability using a quality perspective," Research Report 03A43, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    14. O'Brien, Raymond & Patacchini, Eleonora, 2003. "Testing the exogeneity assumption in panel data models with "non classical" disturbances," Discussion Paper Series In Economics And Econometrics 0302, Economics Division, School of Social Sciences, University of Southampton.
    15. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2002. "Perfection and Stability of Stationary Points with Applications in Noncooperative Games," Discussion Paper 2002-108, Tilburg University, Center for Economic Research.
    16. Edcarlos D. Silva & J. C. Albuquerque & T. R. Cavalcante, 2021. "Fourth-order nonlocal type elliptic problems with indefinite nonlinearities," Partial Differential Equations and Applications, Springer, vol. 2(2), pages 1-22, April.
    17. YongSeog Kim & W. Nick Street & Gary J. Russell & Filippo Menczer, 2005. "Customer Targeting: A Neural Network Approach Guided by Genetic Algorithms," Management Science, INFORMS, vol. 51(2), pages 264-276, February.
    18. Montijano, J.I. & Rández, L. & Van Daele, M. & Calvo, M., 2020. "On the numerical stability of the exponentially fitted methods for first order IVPs," Applied Mathematics and Computation, Elsevier, vol. 379(C).
    19. Yanling Li & Zita Oravecz & Shuai Zhou & Yosef Bodovski & Ian J. Barnett & Guangqing Chi & Yuan Zhou & Naomi P. Friedman & Scott I. Vrieze & Sy-Miin Chow, 2022. "Bayesian Forecasting with a Regime-Switching Zero-Inflated Multilevel Poisson Regression Model: An Application to Adolescent Alcohol Use with Spatial Covariates," Psychometrika, Springer;The Psychometric Society, vol. 87(2), pages 376-402, June.
    20. Jensen, Nathan M. & Li, Quan & Rahman, Aminur, 2007. "Heard melodies are sweet, but those unheard are sweeter : understanding corruption using cross-national firm-level surveys," Policy Research Working Paper Series 4413, The World Bank.

    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:annopr:v:274:y:2019:i:1:d:10.1007_s10479-018-2926-8. 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.