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A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints

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  • Yu-xin Li
  • Jie Zhang
  • Zun-quan Xia

Abstract

To reflect uncertain data in practical problems, stochastic versions of the mathematical program with complementarity constraints (MPCC) have drawn much attention in the recent literature. Our concern is the detailed analysis of convergence properties of a regularization sample average approximation (SAA) method for solving a stochastic mathematical program with complementarity constraints (SMPCC). The analysis of this regularization method is carried out in three steps: First, the almost sure convergence of optimal solutions of the regularized SAA problem to that of the true problem is established by the notion of epiconvergence in variational analysis. Second, under MPCC‐MFCQ, which is weaker than MPCC‐LICQ, we show that any accumulation point of Karash‐Kuhn‐Tucker points of the regularized SAA problem is almost surely a kind of stationary point of SMPCC as the sample size tends to infinity. Finally, some numerical results are reported to show the efficiency of the method proposed.

Suggested Citation

  • Yu-xin Li & Jie Zhang & Zun-quan Xia, 2014. "A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnljam:v:2014:y:2014:i:1:n:321010
    DOI: 10.1155/2014/321010
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    1. 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 with equilibrium constraints," Transportation Research Part B: Methodological, Elsevier, vol. 42(10), pages 843-860, December.
    2. 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.
    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. N. D. Yen, 1997. "Stability of the Solution Set of Perturbed Nonsmooth Inequality Systems and Application," Journal of Optimization Theory and Applications, Springer, vol. 93(1), pages 199-225, April.
    6. 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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