Abstract
In this paper, the discrete approximation of two-stage stochastic variational inequalities has been investigated when the second stage problem has multiple solutions. First, a discrete approximation scheme is given by a series of models with the aid of merit functions. After that, the convergence relationships between these models are analysed, which therefore yields the convergence guarantee of the proposed discrete approximation scheme. Finally, we use the well-known progressive hedging algorithm to report some numerical results and to validate the effectiveness of the discrete approximation approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The authors confirm that all data generated or analyzed during this study are included in this manuscript.
References
Rockafellar, R.T., Wets, R.J.-B.: Stochastic variational inequalities: single-stage to multistage. Math. Progr. 165(1), 331–360 (2017)
Chen, X., Pong, T.K., Wets, R.J.-B.: Two-stage stochastic variational inequalities: an ERM-solution procedure. Math. Progr. 165(1), 71–111 (2017)
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)
Jiang, J., Shi, Y., Wang, X., Chen, X.: Regularized two-stage stochastic variational inequalities for Cournot-Nash equilibrium under uncertainty. J. Comput. Math. 37(6), 813–842 (2019)
Li, M., Zhang, C.: Two-stage stochastic variational inequality arising from stochastic programming. J. Optim. Theory Appl. 186(1), 324–343 (2020)
Wang, X., Chen, X.: Solving two-stage stochastic variational inequalities by a projection semismooth Newton algorithm. Manuscript (2022)
Rockafellar, R.T., Sun, J.: Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging. Math. Progr. 174(1–2), 453–471 (2019)
Chen, X., Sun, H., Xu, H.: Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems. Math. Progr. 177(1–2), 255–289 (2019)
Chen, X., Shapiro, A., Sun, H.: Convergence analysis of sample average approximation of two-stage stochastic generalized equations. SIAM J. Optim. 29(1), 135–161 (2019)
Jiang, J., Chen, X., Chen, Z.: Quantitative analysis for a class of two-stage stochastic linear variational inequality problems. Comput. Optim. Appl. 76(2), 431–460 (2020)
Jiang, J., Li, S.: Regularized sample average approximation approach for two-stage stochastic variational inequalities. J. Optim. Theory Appl. 190(2), 650–671 (2021)
Sun, H., Chen, X.: Two-stage stochastic variational inequalities: Theory, algorithms and applications. J. Oper. Res. Soc. China 9, 1–32 (2021)
Xu, H.: Sample average approximation methods for a class of stochastic variational inequality problems. Asia-Pac. J. Oper. Res. 27(01), 103–119 (2010)
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2007)
Chen, X., Qi, L., Sun, D.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. Comput. 67(222), 519–540 (1998)
Chen, X.: Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equations. J. Comput. Appl. Math. 80(1), 105–126 (1997)
Fukushima, M.: Merit functions for variational inequality and complementarity problems. In: Nonlinear Optimization and Applications, pp. 155–170. Springer, Boston (1996)
Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory, 2nd edn. SIAM, Philadelphia (2014)
Chen, Z., Jiang, J.: Quantitative stability of fully random two-stage stochastic programs with mixed-integer recourse. Optim. Lett. 14(5), 1249–1264 (2020)
Michael, E.: Continuous selections: I. Ann. Math. 63(2), 361–382 (1956)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings, vol. 543. Springer, Berlin (2009)
Han, S., Pang, J.-S.: Continuous selections of solutions to parametric variational inequalities. Manuscript (2022)
Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities: A Qualitative Study. Springer, New York (2005)
Robinson, S.M.: In: König, H., Korte, B., Ritter, K. (eds.) Some continuity properties of polyhedral multifunctions. pp. 206–214. Springer, Berlin, Heidelberg (1981)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (2009)
Acknowledgements
This research is supported by the National Natural Science Foundation of China (12122108, 12261160365 and 12201084) and Postdoctoral Research Foundation of China (2020M673117).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Jiang, J., Sun, H. Discrete approximation for two-stage stochastic variational inequalities. J Glob Optim 89, 117–142 (2024). https://doi.org/10.1007/s10898-023-01337-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-023-01337-1