IDEAS home Printed from https://ideas.repec.org/p/ems/eureri/1164.html
   My bibliography  Save this paper

Simulation-based solution of stochastic mathematical programs with complementarity constraints: Sample-path analysis

Author

Listed:
  • Birbil, S.I.
  • Gürkan, G.
  • Listeş, O.

Abstract

We consider a class of stochastic mathematical programs with complementarity constraints, in which both the objective and the constraints involve limit functions or expectations that need to be estimated or approximated. Such programs can be used for modeling \\average" or steady-state behavior of complex stochastic systems. Recently, simulation-based methods have been successfully used for solving challenging stochastic optimization problems and equilibrium models. Here we broaden the applicability of so-called the sample-path method to include the solution of certain stochastic mathematical programs with equilibrium constraints. The convergence analysis of sample-path methods rely heavily on stability conditions. We first review necessary sensitivity results, then describe the method, and provide sufficient conditions for its almost-sure convergence. Alongside we provide a complementary sensitivity result for the corresponding deterministic problems. In addition, we also provide a unifying discussion on alternative set of sufficient conditions, derive a complementary result regarding the analysis of stochastic variational inequalities, and prove the equivalence of two different regularity conditions.

Suggested Citation

  • Birbil, S.I. & Gürkan, G. & Listeş, O., 2004. "Simulation-based solution of stochastic mathematical programs with complementarity constraints: Sample-path analysis," ERIM Report Series Research in Management ERS-2004-016-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    • Handle: RePEc:ems:eureri:1164
    as

    Download full text from publisher

    File URL: https://repub.eur.nl/pub/1164/ERS%202004%20016%20LIS.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Peter Kall, 1986. "Approximation to Optimization Problems: An Elementary Review," Mathematics of Operations Research, INFORMS, vol. 11(1), pages 9-18, February.
    3. Stephen M. Robinson, 1996. "Analysis of Sample-Path Optimization," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 513-528, August.
    4. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
    5. Ilker Birbil, S. & Gürkan, G. & Listes, O.L., 2004. "Simulation-Based Solution of Stochastic Mathematical Programs with Complementarity Constraints : Sample-Path Analysis," Discussion Paper 2004-25, Tilburg University, Center for Economic Research.
    6. Gurkan, G. & Listes, O.L., 2004. "Simulation-based solution of stochastic mathematical programs with complementarity constraints: sample-path analyis," Econometric Institute Research Papers EI 2004-03, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    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. Ilker Birbil, S. & Gürkan, G. & Listes, O.L., 2004. "Simulation-Based Solution of Stochastic Mathematical Programs with Complementarity Constraints : Sample-Path Analysis," Other publications TiSEM ab097838-6408-404d-aaeb-f, Tilburg University, School of Economics and Management.
    2. Huifu Xu & Fanwen Meng, 2007. "Convergence Analysis of Sample Average Approximation Methods for a Class of Stochastic Mathematical Programs with Equality Constraints," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 648-668, August.
    3. Ş. İlker Birbil & Gül Gürkan & Ovidiu Listeş, 2006. "Solving Stochastic Mathematical Programs with Complementarity Constraints Using Simulation," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 739-760, November.

    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. Ilker Birbil, S. & Gürkan, G. & Listes, O.L., 2004. "Simulation-Based Solution of Stochastic Mathematical Programs with Complementarity Constraints : Sample-Path Analysis," Other publications TiSEM ab097838-6408-404d-aaeb-f, Tilburg University, School of Economics and Management.
    2. Ş. İlker Birbil & Gül Gürkan & Ovidiu Listeş, 2006. "Solving Stochastic Mathematical Programs with Complementarity Constraints Using Simulation," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 739-760, November.
    3. Gurkan, G. & Listes, O.L., 2004. "Simulation-based solution of stochastic mathematical programs with complementarity constraints: sample-path analyis," Econometric Institute Research Papers EI 2004-03, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    4. Ilker Birbil, S. & Gürkan, G. & Listes, O.L., 2004. "Simulation-Based Solution of Stochastic Mathematical Programs with Complementarity Constraints : Sample-Path Analysis," Discussion Paper 2004-25, Tilburg University, Center for Economic Research.
    5. Gürkan, G. & Ozge, A.Y. & Robinson, S.M., 1997. "Sample-path solutions for simulation optimization problems and stochastic variational inequalities," Discussion Paper 1997-78, Tilburg University, Center for Economic Research.
    6. Huifu Xu & Fanwen Meng, 2007. "Convergence Analysis of Sample Average Approximation Methods for a Class of Stochastic Mathematical Programs with Equality Constraints," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 648-668, August.
    7. Patriksson, Michael, 2008. "On the applicability and solution of bilevel optimization models in transportation science: A study on the existence, stability and computation of optimal solutions to stochastic mathematical programs," Transportation Research Part B: Methodological, Elsevier, vol. 42(10), pages 843-860, December.
    8. Gürkan, G. & Ozge, A.Y. & Robinson, S.M., 1997. "Sample-path solutions for simulation optimization problems and stochastic variational inequalities," Other publications TiSEM 9c36c4c8-25d6-4a86-afbd-2, Tilburg University, School of Economics and Management.
    9. Gürkan, G. & Ozge, A.Y., 1996. "Sample-Path Optimization of Buffer Allocations in a Tandem Queue - Part I : Theoretical Issues," Other publications TiSEM 77da022b-635b-46fd-bf4a-f, Tilburg University, School of Economics and Management.
    10. 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.
    11. Daniel Ralph & Huifu Xu, 2011. "Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 568-592, August.
    12. Gürkan, G. & Pang, J.S., 2009. "Approximizations of Nash equilibria," Other publications TiSEM de211d31-d77d-4211-9ca8-2, Tilburg University, School of Economics and Management.
    13. Michael Patriksson & R. Tyrrell Rockafellar, 2002. "A Mathematical Model and Descent Algorithm for Bilevel Traffic Management," Transportation Science, INFORMS, vol. 36(3), pages 271-291, August.
    14. Gürkan, G. & Ozge, A.Y., 1996. "Sample-Path Optimization of Buffer Allocations in a Tandem Queue - Part I : Theoretical Issues," Discussion Paper 1996-98, Tilburg University, Center for Economic Research.
    15. J. S. Pang, 2007. "Partially B-Regular Optimization and Equilibrium Problems," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 687-699, August.
    16. C. Cromvik & M. Patriksson, 2010. "On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 1: Theory," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 461-478, March.
    17. M. Durea & R. Strugariu, 2011. "On parametric vector optimization via metric regularity of constraint systems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(3), pages 409-425, December.
    18. Yongchao Liu & Huifu Xu & Jane J. Ye, 2011. "Penalized Sample Average Approximation Methods for Stochastic Mathematical Programs with Complementarity Constraints," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 670-694, November.
    19. Stefan Scholtes, 2004. "Nonconvex Structures in Nonlinear Programming," Operations Research, INFORMS, vol. 52(3), pages 368-383, June.
    20. Stein, Oliver, 2012. "How to solve a semi-infinite optimization problem," European Journal of Operational Research, Elsevier, vol. 223(2), pages 312-320.

    More about this item

    Keywords

    mathematical programs with equilibrium constraints; regularity conditions; sample-path methods; simulation; stability; stochastic mathematical programs with complementarity constraints;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • M - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics
    • M11 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Administration - - - Production Management
    • R4 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - Transportation Economics

    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:ems:eureri:1164. 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: RePub (email available below). General contact details of provider: https://edirc.repec.org/data/erimanl.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.