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

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  • 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
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    References listed on IDEAS

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    1. Peter Kall, 1986. "Approximation to Optimization Problems: An Elementary Review," Mathematics of Operations Research, INFORMS, vol. 11(1), pages 9-18, February.
    2. 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.
    3. 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.
    4. Stephen M. Robinson, 1996. "Analysis of Sample-Path Optimization," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 513-528, August.
    5. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
    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.
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    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.

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    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

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