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Expected Residual Minimization Formulation for a Class of Stochastic Vector Variational Inequalities

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Abstract

This paper considers a class of vector variational inequalities. First, we present an equivalent formulation, which is a scalar variational inequality, for the deterministic vector variational inequality. Then we concentrate on the stochastic circumstance. By noting that the stochastic vector variational inequality may not have a solution feasible for all realizations of the random variable in general, for tractability, we employ the expected residual minimization approach, which aims at minimizing the expected residual of the so-called regularized gap function. We investigate the properties of the expected residual minimization problem, and furthermore, we propose a sample average approximation method for solving the expected residual minimization problem. Comprehensive convergence analysis for the approximation approach is established as well.

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References

  1. Giannessi, F.: Theorems of alternative, quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, New York (1980)

    Google Scholar 

  2. Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria: Mathematical Theories. Vol. 38, Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  3. Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-Valued and Variational Analysis, vol. 541. Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (2005)

  4. Yang, X.Q., Yao, J.C.: Gap functions and existence of solutions to set-valued vector variational inequalities. J. Optim. Theory Appl. 115, 407–417 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Yang, X.M., Yang, X.Q., Teo, K.L.: Some remarks on the Minty vector variational inequality. J. Optim. Theory Appl. 121, 193–201 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Lee, G.M., Kim, D.S., Lee, B.S., Yen, N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. 34, 745–765 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, G.Y., Goh, C.J., Yang, X.Q.: On gap functions for vector variational inequalities. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories. Nonconvex Optimization and Its Applications, pp. 55–72. Kluwer Academic Publishers, Dordrecht (2000)

    Chapter  Google Scholar 

  8. Li, S.J., Chen, G.Y.: Properties of gap function for vector variational inequality. In: Giannessi, F., Maugeri, A. (eds.) Variational Analysis and Applications. Nonconvex Optimization and Its Applications, pp. 605–631. Springer, Berlin (2005)

    Chapter  Google Scholar 

  9. Li, S.J., Yan, H., Chen, G.Y.: Differential and sensitivity properties of gap functions for vector variational inequalities. Math. Methods Oper. Res. 57, 377–391 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Huang, N.J., Li, J., Yang, X.Q.: Weak sharpness for gap functions in vector variational inequalities. J. Math. Anal. Appl. 394, 449–457 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Charitha, C., Dutta, J.: Regularized gap functions and error bounds for vector variational inequalities. Pac. J. Optim. 6, 497–510 (2010)

    MATH  MathSciNet  Google Scholar 

  12. Charitha, C., Dutta, J., Lalitha, C.S.: Gap functions for vector variational inequalities. Optimization 64, 1499–1520 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ma, H.Q., Huang, N.J., Wu, M., Regan, D.O.: A new gap function for vector variational inequalities with an application. J. Appl. Math. Article ID 423040 (2013)

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Jiang, H.Y., Xu, H.F.: Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Trans. Autom. Control. 53, 1462–1475 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  17. Xu, H.F.: Sample average approximation methods for a class of stochastic variational inequality problems. Asia Pac J. Oper. Res. 27, 103–119 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Lin, G.H., Chen, X.J., Fukushima, M.: New restricted NCP functions and their applications to stochastic NCP and stochastic MPEC. Optimization 56, 641–653 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Zhang, Y.F., Chen, X.J.: Regularizations for stochastic linear variational inequalities. J. Optim. Theory Appl. 163, 460–481 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  23. Luo, M.J., Lin, G.H.: Expected residual minimization method for stochastic variational inequality problems. J. Optim. Theory Appl. 140, 103–116 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Luo, M.J., Lin, G.H.: Convergence results of the ERM method for nonlinear stochastic variational inequality problems. J. Optim. Theory Appl. 142, 569–581 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Luo, M.J., Lin, G.H.: Stochastic variational inequality problems with additional constraints and their applications in supply chain network equilibria. Pac. J. Optim. 7, 263–279 (2011)

    MATH  MathSciNet  Google Scholar 

  26. Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  27. Patrick, B.: Probability and Measure. Wiley, New York (1995)

    MATH  Google Scholar 

  28. Shapiro, A., Xu, H.F.: Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation. Optimization. 57, 395–418 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Andreani, R., Martinez, J.M., Ramos, A., Silva, P.J.S.: A cone-continuity constraint qualification and algorithmic consequences. SIAM J. Optim. 26, 96–110 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  30. Andreani, R., Haeser, G., Martinez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization 60, 627–641 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  31. Ralph, D., Xu, H.F.: Convergence of stationary points of sample average two-stage stochastic programs: a generalized equation approach. Math. Oper. Res. 36, 568–592 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This work was supported in part by the NSFC Grant (No. 11431004), the Hongkong Baptist University FRG1/15-16/027 and RC-NACAN-ZHANG J, the Humanity and Social Science Foundation of Ministry of Education of China (No. 15YJA630034) and the Innovation Program of Shanghai Municipal Education Commission (No. 14ZS086). The authors are grateful to Professor F. Giannessi, Professor G.Y. Chen, and two anonymous referees for their valuable comments and constructive suggestions, which help to improve the original manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, 610065, China

    Yong Zhao

  2. Department of Mathematics, Hong Kong Baptist University, Hongkong, China

    Jin Zhang

  3. Department of Mathematics, Chongqing Normal University, Chongqing, 401131, China

    Xinmin Yang

  4. School of Management, Shanghai University, Shanghai, 200444, China

    Gui-Hua Lin

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  1. Yong Zhao
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  2. Jin Zhang
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  3. Xinmin Yang
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  4. Gui-Hua Lin
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Correspondence to Yong Zhao.

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Communicated by Guang-ya Chen.

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Zhao, Y., Zhang, J., Yang, X. et al. Expected Residual Minimization Formulation for a Class of Stochastic Vector Variational Inequalities. J Optim Theory Appl 175, 545–566 (2017). https://doi.org/10.1007/s10957-016-0939-5

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  • DOI: https://doi.org/10.1007/s10957-016-0939-5

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