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Stochastic Programming with Equilibrium Constraints

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Abstract

In this paper, we discuss here-and-now type stochastic programs with equilibrium constraints. We give a general formulation of such problems and study their basic properties such as measurability and continuity of the corresponding integrand functions. We discuss also the consistency and rate of convergence of sample average approximations of such stochastic problems

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References

  1. J.R. Birge F. Louveaux (1997) Introduction to Stochastic Programming Springer New York, NY

    Google Scholar 

  2. M. Patriksson L. Wynter (1999) ArticleTitleStochastic Mathematical Programs with Equilibrium Constraints Operations Research Letters. 25 159–167 Occurrence Handle10.1016/S0167-6377(99)00052-8 Occurrence Handle2001b:90053

    Article  MathSciNet  Google Scholar 

  3. A. Evgrafov M. Patriksson (2004) ArticleTitleOn the Existence of Solutions to Stochastic Mathematical Programs with Equilibrium Constraints Journal of Optimization Theory and Applications. 121 65–76 Occurrence Handle10.1023/B:JOTA.0000026131.04418.b7 Occurrence Handle2005c:90057

    Article  MathSciNet  Google Scholar 

  4. G.H. Lin M. Fukushima (2003) A Class of Mathematical Programs with Complementarity Constraints: Reformulations and Algorithms Department of Applied Mathematics and Physics, Kyoto University Kyoto, Japan

    Google Scholar 

  5. S.I. Birbil G. Gürkan O. Listes (2004) Simulation-Based Solution of Stochastic Mathematical Programs with Complementarity Constraints: Sample-Path Analysis Erasmus University Rotterdam Holland

    Google Scholar 

  6. Xu, H., An Approach for Stochastic Stackelberg-Nash-Cournot Equilibrium, Optimization (to appear).

  7. Xu, H., An Implicit Programming Approach for a class of Stochastic Mathematics Programs with Complementarity Constraints, Preprint, 2005.

  8. Z.Q. Luo J.S. Pang D. Ralph (1996) Mathematical Programs with Equilibrium Constraints Cambridge University Press Cambridge, UK

    Google Scholar 

  9. H. Scheel S. Scholtes (2000) ArticleTitleMathematical Programs with Complementarity Constraints: Stationarity Optimality and Sensitivity Mathematics of Operations Research. 25 1–22 Occurrence Handle10.1287/moor.25.1.1.15213 Occurrence Handle2002f:90136

    Article  MathSciNet  Google Scholar 

  10. S.M. Robinson (1980) ArticleTitleStrongly Regular Generalized Equations Mathematics of Operations Research. 5 43–62 Occurrence Handle0437.90094 Occurrence Handle81m:90109

    MATH  MathSciNet  Google Scholar 

  11. A. Ruszczyński A. Shapiro (2003) Stochastic Programming Handbooks in Operations Research and Management Science, North-Holland Publishing Company Amsterdam, Holland

    Google Scholar 

  12. G. Castaing M. Valadier (1977) Convex Analysis and Measurable Multifunctions Springer Berlin, Germany

    Google Scholar 

  13. R.T. Rockafellar R.J.B. Wets (1998) Variational Analysis Springer Verlag New York, NY

    Google Scholar 

  14. G. Auchmuty (1989) ArticleTitleVariational Principles for Variational Inequalities Numerical Functional Analysis and Optimization. 10 863–874 Occurrence Handle0678.49010 Occurrence Handle91b:49010

    MATH  MathSciNet  Google Scholar 

  15. M. Fukushima (1992) ArticleTitleEquivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems Mathematical Programming. 53 99–110 Occurrence Handle10.1007/BF01585696 Occurrence Handle0756.90081 Occurrence Handle92k:90100

    Article  MATH  MathSciNet  Google Scholar 

  16. F. Facchinei J.S. Pang (2003) Finite-Dimensional Variational Inequalities and Complementarity Problems Springer New York, NY

    Google Scholar 

  17. J.F. Bonnans A. Shapiro (2000) Perturbation Analysis of Optimization Problems Springer New York, NY

    Google Scholar 

  18. D. Kinderlehrer G. Stampacchia (1980) An Introduction to Variational Inequalities and Their Applications Academic Press New York, NY

    Google Scholar 

  19. A. Shapiro (2005) ArticleTitleSensitivity Analysis of Parametrized Variational Inequalities Mathematics of Operations Research. 30 76–91 Occurrence Handle10.1287/moor.1040.0115

    Article  Google Scholar 

  20. S.M. Robinson (1982) ArticleTitleGeneralized Equations and Their Solutions, Part 2: Applications to Nonlinear Programming Mathematical Programming Study. 19 200–221 Occurrence Handle0495.90077

    MATH  Google Scholar 

  21. X. Hu D. Ralph (1982) ArticleTitleA Note on Sensitivity of Value Functions of Mathematical Programs with Complementarity Constraints Mathematical Programming. 93 265–279 Occurrence Handle2003j:90059

    MathSciNet  Google Scholar 

  22. A. Shapiro (2003) Monte Carlo Sampling Methods Handbooks in Operations Research and Management Science, North-Holland Publishing Company Amsterdam, Holland

    Google Scholar 

  23. Shapiro A., and Nemirovski, A., On Complexity of Stochastic Programming Problems; see http://www.optimization-online.org2004.

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Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia

    A. Shapiro

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  1. A. Shapiro
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Communicated by P. M. Pardalos

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Shapiro, A. Stochastic Programming with Equilibrium Constraints. J Optim Theory Appl 128, 221–243 (2006). https://doi.org/10.1007/s10957-005-7566-x

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  • DOI: https://doi.org/10.1007/s10957-005-7566-x

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