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The deterministic ERM and CVaR reformulation for the stochastic generalized complementarity problem

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Abstract

In this paper, we consider the stochastic generalized complementarity problem (SGCP), which has a variety of important applications and has the stochastic complementarity problem as a special case. In order to give reasonable solutions of SGCP, we propose a deterministic formulation that minimizes the expected value of the residual function as the objective function and uses CVaR constraints to ensure the probabilistic feasibility of its solutions. The resulting model containsnon-smooth constraints and expected value functions both in the objective function and constraints. We present approximation problems for solving this model using smoothing and Monte-Carlo techniques. We prove two convergence results of the proposed approximation problems. One is to increase the sample size only, the other is to let smoothing parameter tend to zero as sample increase together. The effectiveness of the proposed approach was demonstrated using numerical examples.

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References

  1. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Financ. 9, 203–228 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, B.T., Harker, P.T.: Smooth approximations to nonlinear complementarity problems. SIAM J. Optim. 7, 403–420 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, J.X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, J.X., Lin, G.H.: CVaR-based formulation and approximation method for stochastic variational inequalities. Numer. Algebra Control Optim. 1, 35–48 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, J.X., Wets, R.J.B., Zhang, Y.F.: Stochastic variational inequalities: residual minimization smoothing sample average approximations. SIAM J. Optim. 22, 649–673 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, J.X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. Fang, H.T., Chen, X.J., Fukushima, M.: Stochastic \(R_0\) matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hamatani, K., Fukushima, M.: Pricing American options with uncertain volatility through stochastic linear complementarity models. Comput. Optim. Appl. 50, 263–286 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kalos, M.H., Whitlock, P.A.: Monte Carlo Methods. Wiley, New York (1986)

    Book  MATH  Google Scholar 

  12. Kall, P.: Approximation to optimization problems: an elementary review. Math. Oper. Res. 11, 8–17 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Karamardian, S.: Generalized complementarity problem. J. Optim. Theory Appl. 8, 161–167 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ling, C., Qi, L.Q., Zhou, G.L., Caccetta, L.: The \(SC^1\) property of an expected residual function arising from stochastic complementarity problems. Oper. Res. Lett. 36, 456–460 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lin, G.H.: Combined Monte Carlo sampling and penalty method for stochastic nonlinear complementarity problems. Math. Comput. 78, 1671–1686 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, G.H., Fukushima, M.: Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: A survey. Pacific J. Optim. 6, 455–482 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Lin, G.H., Fukushima, M.: New reformulations for stochastic nonlinear complementarity problems. Optim. Methods Softw. 21, 551–564 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17, 969–996 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)

    Article  Google Scholar 

  20. Ruszczyski, A., Shapiro, A.: Handbooks in Operations Research and Management Science, 10: Stochastic Programming. Elsevier, Amsterdam (2003)

    Google Scholar 

  21. Wang, M., Ali, M.M.: Stochastic nonlinear complementarity problems: stochastic programming reformulation and penalty-based approximation method. J. Optim. Theory Appl. 144, 597–614 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xu, L.Y., Yu, B.: CVaR-constrained stochastic programming reformulation for stochastic nonlinear complementarity problems. Comput. Optim. Appl. 58, 483–501 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, C., Chen, X.: Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty. J. Optim. Theory Appl. 137, 277–295 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Liaoning University, Liaoning, 110036, China

    Meiju Luo

  2. Industrial and Manufacturing Systems Engineering, Iowa State University, Ames, 50011, USA

    Lizhi Wang

Authors
  1. Meiju Luo
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  2. Lizhi Wang
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Corresponding author

Correspondence to Meiju Luo.

Additional information

This work was supported in part by National Natural Science Foundation of China (No. 11501275) and Scientific Research Fund of Liaoning Provincial Education Department (No. L2015199).

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Luo, M., Wang, L. The deterministic ERM and CVaR reformulation for the stochastic generalized complementarity problem. Japan J. Indust. Appl. Math. 34, 321–333 (2017). https://doi.org/10.1007/s13160-017-0262-z

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  • DOI: https://doi.org/10.1007/s13160-017-0262-z

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