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On Multistage Pseudomonotone Stochastic Variational Inequalities

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Abstract

This paper focuses on the solvability of multistage pseudomonotone stochastic variational inequalities (SVIs). On the one hand, some known solvability results of pseudomonotone deterministic variational inequalities cannot be directly extended to multistage pseudomonotone SVIs, so we construct the isomorphism between both and then establish theoretical results on the existence, convexity, boundedness and compactness of the solution set for multistage pseudomonotone SVIs via such an isomorphism. On the other hand, there does not exist a special algorithm for solving the multistage pseudomonotone SVIs so far, so we propose some sufficient conditions on the elicitability of multistage pseudomonotone SVIs, which opens the door for applying Rockafellar’s elicited progressive hedging algorithm to solve such SVIs. Numerical results on solving a two-stage stochastic market optimization problem and randomly generated two-stage pseudomonotone linear complementarity problems are presented.

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We do not analyze or generate any datasets, since our work proceeds within a theoretical and mathematical approach.

References

  1. Cambini, A., Martein, L.: Generalized Convexity and Optimization: Theory and Applications. Springer, Berlin (2009)

    MATH  Google Scholar 

  2. Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30(4), 1022–1038 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, X., Pong, T., Wets, R.J.B.: Two-stage stochastic variational inequalities: an ERM-solution procedure. Math. Program. 165(1–2), 71–111 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117(1–2), 51–80 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Eckstein, J., Bertsekas, D.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1–2), 293–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  7. Fang, H., Chen, X., Fukushima, M.: Stochastic \(R_0\) matrix linear complementarity problems. SIAM J. Optim. 18(2), 482–506 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84(1–2), 313–333 (1999)

    MathSciNet  MATH  Google Scholar 

  9. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2013)

  10. Iusem, A., Jofré, A., Oliveira, R.I., Thompson, P.: Extragradient method with variance reduction for stochastic variational inequalities. SIAM J. Optim. 27(2), 686–724 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jiang, H., Xu, H.: Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Trans. Autom. Control 53(6), 1462–1475 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jiang, J., Sun, H.: Monotonicity and complexity of multistage stochastic variational inequalities. J. Optim. Theory Appl. (2022). https://doi.org/10.1007/s10957-022-02099-8

    Article  MATH  Google Scholar 

  13. Jiang, J., Sun, H., Zhou, B.: Convergence analysis of sample average approximation for a class of stochastic nonlinear complementarity problems: from two-stage to multistage. Numer. Algor. 89(1), 167–194 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kannan, A., Shanbhag, U.V.: Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants. Comput. Optim. Appl. 74(3), 779–820 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ling, C., Qi, L., Zhou, G., Caccetta, L.: The SC1 property of an expected residual function arising from stochastic complementarity problems. Oper. Res. Lett. 36(4), 456–460 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luo, M., Lin, G.: Expected residual minimization method for stochastic variational inequality problems. J. Optim. Theory Appl. 140(1), 103–116 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. McCarl, B.A., Moskowitz, H., Furtan, H.: Quadratic programming applications. Int. J. Mgmt. Sci. 5(1), 43–55 (1977)

    Google Scholar 

  18. Pang, J.S., Sen, S., Shanbhag, U.V.: Two-stage non-cooperative games with risk-averse players. Math. Program. 165(1–2), 235–290 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Philippe, G.C.: Linear and Nonlinear Functional Analysis with Applications. SIAM, Philadelphia (2013)

    Google Scholar 

  20. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58(1–2), 353–367 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149(1), 75–88 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control. Optim. 14(5), 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rockafellar, R.T.: Progressive decoupling of linkages in optimization and variational inequalities with elicitable convexity or monotonicity. Set-Valued Var. Anal. 27(4), 863–893 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rockafellar, R.T., Sun, J.: Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging. Math. Program. 174(1–2), 453–471 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rockafellar, R.T., Sun, J.: Solving Lagrangian variational inequalities with applications to stochastic programming. Math. Program. 181(1–2), 435–451 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  27. Rockafellar, R.T., Wets, R.J.B.: Stochastic variational inequalities: single-stage to multistage. Math. Program. 165(1–2), 291–330 (2017)

  28. Sun, J., Zhang, M.: The elicited progressive decoupling algorithm: a note on the rate of convergence and a preliminary numerical experiment on the choice of parameters. Set-Valued Var. Anal. 29(1), 997–1018 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Xu, H.: Sample average approximation methods for a class of variational inequality problems. Asia Pac. J. Oper. Res. 27(1), 103–119 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yousefian, F., Nedić, A., Shanbhag, U.V.: On stochastic mirror-prox algorithms for stochastic Cartesian variational inequalities: randomized block coordinate and optimal averaging schemes. Set-Valued Var. Anal. 26(4), 789–819 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, C., Chen, X.: Smoothing projected gradient method and its application to stochastic linear complementarity problems. SIAM J. Optim. 20(2), 627–649 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, M., Hou, L., Sun, J., Yan, A.: A model of multistage risk-averse stochastic optimization and its solution by scenario-based decomposition algorithms. Asia-Pac. J. Oper. Res. 37(4), 627–649 (2004)

    MathSciNet  Google Scholar 

  33. Zhang, M., Sun, J., Xu, H.: Two-stage quadratic games under uncertainty and their solution by progressive hedging algorithms. SIAM J. Optim. 29(3), 1799–1818 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Liping Zhang’s work was supported by the National Natural Science Foundation of China (Grant No. 12171271).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Xingbang Cui & Liping Zhang

  2. National University of Singapore, Singapore, 119245, Singapore

    Jie Sun

  3. Curtin University, Perth, WA, 6845, Australia

    Jie Sun

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  1. Xingbang Cui
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  2. Jie Sun
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  3. Liping Zhang
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Correspondence to Liping Zhang.

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Communicated by Xinmin Yang.

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Cui, X., Sun, J. & Zhang, L. On Multistage Pseudomonotone Stochastic Variational Inequalities. J Optim Theory Appl 199, 363–391 (2023). https://doi.org/10.1007/s10957-023-02289-y

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