IDEAS home Printed from https://ideas.repec.org/p/ver/wpaper/16-2021.html
   My bibliography  Save this paper

Some numerical aspects on a method for solving linear problems with complementarity constraints

Author

Listed:
  • Giandomenico Mastroeni

    (Department of Computer Science, University of Pisa, Italy)

  • Letizia Pellegrini

    (Department of Economics (University of Verona))

  • Alberto Peretti

    (Department of Economics (University of Verona))

Abstract

A known method for solving linear problems with complementarity constraints is briefly recalled. The method decomposes the given problem in a sequence of parameterized problems and - by means of suitable cuts - allows to define an iterative procedure that leads to an optimal solution or to an approximation of it providing an estimate of the error. In this paper, for problems of different dimensions we have implemented some numerical experiments which show that in most cases the method converges linearly with respect to the dimension of the problem. Our results are also compared with those obtained by similar approaches where different kinds of cuts are considered.

Suggested Citation

  • Giandomenico Mastroeni & Letizia Pellegrini & Alberto Peretti, 2021. "Some numerical aspects on a method for solving linear problems with complementarity constraints," Working Papers 16/2021, University of Verona, Department of Economics.
    • Handle: RePEc:ver:wpaper:16/2021
    as

    Download full text from publisher

    File URL: http://dse.univr.it/home/workingpapers/wp2021n16.pdf
    File Function: First version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Christian Kanzow & Alexandra Schwartz, 2015. "The Price of Inexactness: Convergence Properties of Relaxation Methods for Mathematical Programs with Complementarity Constraints Revisited," Mathematics of Operations Research, INFORMS, vol. 40(2), pages 253-275, February.
    2. Gui-Hua Lin & Masao Fukushima, 2005. "A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints," Annals of Operations Research, Springer, vol. 133(1), pages 63-84, January.
    3. X. M. Hu & D. Ralph, 2004. "Convergence of a Penalty Method for Mathematical Programming with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 365-390, November.
    4. Alberto Ramos, 2019. "Two New Weak Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and Applications," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 566-591, November.
    5. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, February.
    6. Mounir Haddou & Patrick Maheux, 2014. "Smoothing Methods for Nonlinear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 711-729, March.
    7. Yu Chen & Zhong Wan, 2018. "A New Smoothing Method for Mathematical Programs with Complementarity Constraints Based on Logarithm-Exponential Function," Mathematical Problems in Engineering, Hindawi, vol. 2018, pages 1-11, August.
    8. Xu, Huayu & Pang, Jong-Shi & Ordóñez, Fernando & Dessouky, Maged, 2015. "Complementarity models for traffic equilibrium with ridesharing," Transportation Research Part B: Methodological, Elsevier, vol. 81(P1), pages 161-182.
    9. 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.
    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. Christian Kanzow & Alexandra Schwartz, 2015. "The Price of Inexactness: Convergence Properties of Relaxation Methods for Mathematical Programs with Complementarity Constraints Revisited," Mathematics of Operations Research, INFORMS, vol. 40(2), pages 253-275, February.
    2. H. Z. Luo & X. L. Sun & Y. F. Xu, 2010. "Convergence Properties of Modified and Partially-Augmented Lagrangian Methods for Mathematical Programs with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 145(3), pages 489-506, June.
    3. Jean-Pierre Dussault & Mounir Haddou & Abdeslam Kadrani & Tangi Migot, 2020. "On Approximate Stationary Points of the Regularized Mathematical Program with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 504-522, August.
    4. Hoai An Thi & Thi Minh Tam Nguyen & Tao Pham Dinh, 2023. "On solving difference of convex functions programs with linear complementarity constraints," Computational Optimization and Applications, Springer, vol. 86(1), pages 163-197, September.
    5. Suhong Jiang & Jin Zhang & Caihua Chen & Guihua Lin, 2018. "Smoothing partial exact penalty splitting method for mathematical programs with equilibrium constraints," Journal of Global Optimization, Springer, vol. 70(1), pages 223-236, January.
    6. Lei Guo & Gui-Hua Lin & Jane J. Ye, 2013. "Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 33-64, July.
    7. Gui-Hua Lin & Masao Fukushima, 2005. "A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints," Annals of Operations Research, Springer, vol. 133(1), pages 63-84, January.
    8. Na Xu & Xide Zhu & Li-Ping Pang & Jian Lv, 2018. "Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(01), pages 1-20, February.
    9. S A Gabriel & Y Shim & A J Conejo & S de la Torre & R García-Bertrand, 2010. "A Benders decomposition method for discretely-constrained mathematical programs with equilibrium constraints," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(9), pages 1404-1419, September.
    10. Filippo Pecci & Edo Abraham & Ivan Stoianov, 2017. "Penalty and relaxation methods for the optimal placement and operation of control valves in water supply networks," Computational Optimization and Applications, Springer, vol. 67(1), pages 201-223, May.
    11. Winterfeld, Anton, 2008. "Application of general semi-infinite programming to lapidary cutting problems," European Journal of Operational Research, Elsevier, vol. 191(3), pages 838-854, December.
    12. Daniel Ralph & Oliver Stein, 2011. "The C-Index: A New Stability Concept for Quadratic Programs with Complementarity Constraints," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 504-526, August.
    13. Martin Branda & Max Bucher & Michal Červinka & Alexandra Schwartz, 2018. "Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization," Computational Optimization and Applications, Springer, vol. 70(2), pages 503-530, June.
    14. Hu, X. & Ralph, R., 2006. "Using EPECs to model bilevel games in restructured electricity markets with locational prices," Cambridge Working Papers in Economics 0619, Faculty of Economics, University of Cambridge.
    15. H. Luo & X. Sun & Y. Xu & H. Wu, 2010. "On the convergence properties of modified augmented Lagrangian methods for mathematical programming with complementarity constraints," Journal of Global Optimization, Springer, vol. 46(2), pages 217-232, February.
    16. Meng Xu & Guangmin Wang & Susan Grant-Muller & Ziyou Gao, 2017. "Joint road toll pricing and capacity development in discrete transport network design problem," Transportation, Springer, vol. 44(4), pages 731-752, July.
    17. Lei Guo & Gui-Hua Lin, 2013. "Notes on Some Constraint Qualifications for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 600-616, March.
    18. Jian Yao & Ilan Adler & Shmuel S. Oren, 2008. "Modeling and Computing Two-Settlement Oligopolistic Equilibrium in a Congested Electricity Network," Operations Research, INFORMS, vol. 56(1), pages 34-47, February.
    19. J. J. Júdice & H. D. Sherali & I. M. Ribeiro & A. M. Faustino, 2007. "Complementarity Active-Set Algorithm for Mathematical Programming Problems with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 467-481, September.
    20. Joaquim Júdice, 2012. "Algorithms for linear programming with linear complementarity constraints," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(1), pages 4-25, April.

    More about this item

    Keywords

    Mathematical programs with complementarity constraints; duality; decomposition methods;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

    Statistics

    Access and download statistics

    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:ver:wpaper:16/2021. 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: Michael Reiter (email available below). General contact details of provider: https://edirc.repec.org/data/isverit.html .

    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.