# American Institute of Mathematical Sciences

• Previous Article
A full-modified-Newton step $O(n)$ infeasible interior-point method for the special weighted linear complementarity problem
• JIMO Home
• This Issue
• Next Article
An efficient iterative method for solving split variational inclusion problem with applications
doi: 10.3934/jimo.2021083
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Quantitative stability of the ERM formulation for a class of stochastic linear variational inequalities

 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: Shengjie Li

Received  November 2020 Revised  March 2021 Early access April 2021

This paper focuses on the quantitative stability analysis of the expected residual minimization (ERM) formulation for a class of stochastic linear variational inequalities. Firstly, the existence of solutions of the ERM formulation and its perturbed problem is discussed. Then, the quantitative stability of the ERM formulation is derived under suitable probability metrics. Finally, the sample average approximation (SAA) problem of the ERM formulation is studied, and the rates of convergence of optimal solution sets are obtained under different assumptions.

Citation: Jianxun Liu, Shengjie Li, Yingrang Xu. Quantitative stability of the ERM formulation for a class of stochastic linear variational inequalities. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021083
##### References:
 [1] R. P. Agdeppa, N. Yamashita and M. Fukushima, Convex expected residual models for stochastic affine variational inequality problems and its application to the traffic equilibrium problem, Pac. J. Optim., 6 (2010), 3-19.   Google Scholar [2] X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Math. Oper. Res., 30 (2005), 1022-1038.  doi: 10.1287/moor.1050.0160.  Google Scholar [3] X. Chen, R. J.-B. Wets and Y. Zhang, Stochastic variational inequalities: Residual minimization smoothing sample average approximations, SIAM J. Optim., 22 (2012), 649-673.  doi: 10.1137/110825248.  Google Scholar [4] F. Facchinei and J. -S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003.  Google Scholar [5] H. Fang, X. Chen and M. Fukushima, Stochastic $R_{0}$ matrix linear complementarity problems, SIAM J. Optim., 18 (2007), 482-506.  doi: 10.1137/050630805.  Google Scholar [6] J. Jiang and Z. Chen, Quantitative stability of multistage stochastic programs via calm modifications, Oper. Res. Lett., 46 (2018), 543-547.  doi: 10.1016/j.orl.2018.08.007.  Google Scholar [7] J. Jiang and Z. Chen, Quantitative stability analysis of two-stage stochastic linear programs with full random recourse, Numer. Func. Anal. Optim., 40 (2019), 1847-1876.  doi: 10.1080/01630563.2019.1639729.  Google Scholar [8] J. Jiang, X. Chen and Z. Chen, Quantitative analysis for a class of two-stage stochastic linear variational inequality problems, Comput. Optim. Appl., 76 (2020), 431-460.  doi: 10.1007/s10589-020-00185-z.  Google Scholar [9] J. Jiang, Z. Chen and X. M. Yang, Rates of convergence of sample average approximation under heavy tailed distributions, To preprint on Optimization Online. Google Scholar [10] Z. Lin and Z. Bai, Probability Inequalities, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-05261-3.  Google Scholar [11] J. Liu and S. Li, Unconstrained optimization reformulation for stochastic nonlinear complementarity problems, Appl. Anal., (2019). doi: 10.1080/00036811.2019.1636969.  Google Scholar [12] F. Lu and S.-J. Li, Method of weighted expected residual for solving stochastic variational inequality problems, Appl. Math. Comput., 269 (2015), 651-663.  doi: 10.1016/j.amc.2015.07.115.  Google Scholar [13] M. J. Luo and G. H. Lin, Expected residual minimization method for stochastic variation inequality problems, J. Optim. Theory Appl., 140 (2009), 103-116.  doi: 10.1007/s10957-008-9439-6.  Google Scholar [14] M. J. Luo and G. H. Lin, Convergence results of the ERM method for nonlinear stochastic variation inequality problems, J. Optim. Theory. Appl., 142 (2009), 569-581.  doi: 10.1007/s10957-009-9534-3.  Google Scholar [15] L. Mirsky, Symmetric gauge functions and unitarily invariant norms, Quart J. Math. Oxford, 11 (1960), 50-59.  doi: 10.1093/qmath/11.1.50.  Google Scholar [16] S. T. Rachev, L. B. Klebanov, S. V. Stoyanov and F. J. Fobozzi, The Methods of Distances in the Theory of Probility and Statistics, Springer, New York, 2013. doi: 10.1007/978-1-4614-4869-3.  Google Scholar [17] S. T. Rachev and W. Römisch, Quantitative stability in stochastic programming: The method of probability metrics, Math. Oper. Res., 27 (2002), 792-818.  doi: 10.1287/moor.27.4.792.304.  Google Scholar [18] W. Römisch, Stability of stochastic programming problems, Stochastic programming, 10 (2003), 483-554.  doi: 10.1016/S0927-0507(03)10008-4.  Google Scholar [19] A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, 2014. doi: 10.1137/1.9780898718751.  Google Scholar [20] A. Shapiro and H. Xu, Stochastic mathematical programs with with equilibrium constraints, modelling and sample average approximation, Optimization, 57 (2008), 395-418.  doi: 10.1080/02331930801954177.  Google Scholar [21] H. Xu, Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming, J. Math. Anal. Appl., 368 (2010), 692-710.  doi: 10.1016/j.jmaa.2010.03.021.  Google Scholar [22] H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems, Asia-Pac. J. Oper. Res., 27 (2010), 103-119.  doi: 10.1142/S0217595910002569.  Google Scholar [23] Y. Zhang and X. Chen, Regularizations for stochastic linear variational inequalities, J. Optim. Theory Appl., 163 (2014), 460-481.  doi: 10.1007/s10957-013-0514-2.  Google Scholar [24] Y. Zhao, J. Zhang, X. Yang and G.-H. Lin, Expected residual minimization formulation for a class of stochastic vector variational inequalities, J. Optim. Theory Appl., 175 (2017), 545-566.  doi: 10.1007/s10957-016-0939-5.  Google Scholar

show all references

##### References:
 [1] R. P. Agdeppa, N. Yamashita and M. Fukushima, Convex expected residual models for stochastic affine variational inequality problems and its application to the traffic equilibrium problem, Pac. J. Optim., 6 (2010), 3-19.   Google Scholar [2] X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Math. Oper. Res., 30 (2005), 1022-1038.  doi: 10.1287/moor.1050.0160.  Google Scholar [3] X. Chen, R. J.-B. Wets and Y. Zhang, Stochastic variational inequalities: Residual minimization smoothing sample average approximations, SIAM J. Optim., 22 (2012), 649-673.  doi: 10.1137/110825248.  Google Scholar [4] F. Facchinei and J. -S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003.  Google Scholar [5] H. Fang, X. Chen and M. Fukushima, Stochastic $R_{0}$ matrix linear complementarity problems, SIAM J. Optim., 18 (2007), 482-506.  doi: 10.1137/050630805.  Google Scholar [6] J. Jiang and Z. Chen, Quantitative stability of multistage stochastic programs via calm modifications, Oper. Res. Lett., 46 (2018), 543-547.  doi: 10.1016/j.orl.2018.08.007.  Google Scholar [7] J. Jiang and Z. Chen, Quantitative stability analysis of two-stage stochastic linear programs with full random recourse, Numer. Func. Anal. Optim., 40 (2019), 1847-1876.  doi: 10.1080/01630563.2019.1639729.  Google Scholar [8] J. Jiang, X. Chen and Z. Chen, Quantitative analysis for a class of two-stage stochastic linear variational inequality problems, Comput. Optim. Appl., 76 (2020), 431-460.  doi: 10.1007/s10589-020-00185-z.  Google Scholar [9] J. Jiang, Z. Chen and X. M. Yang, Rates of convergence of sample average approximation under heavy tailed distributions, To preprint on Optimization Online. Google Scholar [10] Z. Lin and Z. Bai, Probability Inequalities, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-05261-3.  Google Scholar [11] J. Liu and S. Li, Unconstrained optimization reformulation for stochastic nonlinear complementarity problems, Appl. Anal., (2019). doi: 10.1080/00036811.2019.1636969.  Google Scholar [12] F. Lu and S.-J. Li, Method of weighted expected residual for solving stochastic variational inequality problems, Appl. Math. Comput., 269 (2015), 651-663.  doi: 10.1016/j.amc.2015.07.115.  Google Scholar [13] M. J. Luo and G. H. Lin, Expected residual minimization method for stochastic variation inequality problems, J. Optim. Theory Appl., 140 (2009), 103-116.  doi: 10.1007/s10957-008-9439-6.  Google Scholar [14] M. J. Luo and G. H. Lin, Convergence results of the ERM method for nonlinear stochastic variation inequality problems, J. Optim. Theory. Appl., 142 (2009), 569-581.  doi: 10.1007/s10957-009-9534-3.  Google Scholar [15] L. Mirsky, Symmetric gauge functions and unitarily invariant norms, Quart J. Math. Oxford, 11 (1960), 50-59.  doi: 10.1093/qmath/11.1.50.  Google Scholar [16] S. T. Rachev, L. B. Klebanov, S. V. Stoyanov and F. J. Fobozzi, The Methods of Distances in the Theory of Probility and Statistics, Springer, New York, 2013. doi: 10.1007/978-1-4614-4869-3.  Google Scholar [17] S. T. Rachev and W. Römisch, Quantitative stability in stochastic programming: The method of probability metrics, Math. Oper. Res., 27 (2002), 792-818.  doi: 10.1287/moor.27.4.792.304.  Google Scholar [18] W. Römisch, Stability of stochastic programming problems, Stochastic programming, 10 (2003), 483-554.  doi: 10.1016/S0927-0507(03)10008-4.  Google Scholar [19] A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, 2014. doi: 10.1137/1.9780898718751.  Google Scholar [20] A. Shapiro and H. Xu, Stochastic mathematical programs with with equilibrium constraints, modelling and sample average approximation, Optimization, 57 (2008), 395-418.  doi: 10.1080/02331930801954177.  Google Scholar [21] H. Xu, Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming, J. Math. Anal. Appl., 368 (2010), 692-710.  doi: 10.1016/j.jmaa.2010.03.021.  Google Scholar [22] H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems, Asia-Pac. J. Oper. Res., 27 (2010), 103-119.  doi: 10.1142/S0217595910002569.  Google Scholar [23] Y. Zhang and X. Chen, Regularizations for stochastic linear variational inequalities, J. Optim. Theory Appl., 163 (2014), 460-481.  doi: 10.1007/s10957-013-0514-2.  Google Scholar [24] Y. Zhao, J. Zhang, X. Yang and G.-H. Lin, Expected residual minimization formulation for a class of stochastic vector variational inequalities, J. Optim. Theory Appl., 175 (2017), 545-566.  doi: 10.1007/s10957-016-0939-5.  Google Scholar
 [1] Suxiang He, Pan Zhang, Xiao Hu, Rong Hu. A sample average approximation method based on a D-gap function for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 977-987. doi: 10.3934/jimo.2014.10.977 [2] Mei Ju Luo, Yi Zeng Chen. Smoothing and sample average approximation methods for solving stochastic generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 1-15. doi: 10.3934/jimo.2016.12.1 [3] Mingzheng Wang, M. Montaz Ali, Guihua Lin. Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks. Journal of Industrial & Management Optimization, 2011, 7 (2) : 317-345. doi: 10.3934/jimo.2011.7.317 [4] Xiaojun Chen, Guihua Lin. CVaR-based formulation and approximation method for stochastic variational inequalities. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35 [5] S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155 [6] Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645 [7] Yongchao Liu. Quantitative stability analysis of stochastic mathematical programs with vertical complementarity constraints. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 451-460. doi: 10.3934/naco.2018028 [8] Liping Pang, Fanyun Meng, Jinhe Wang. Asymptotic convergence of stationary points of stochastic multiobjective programs with parametric variational inequality constraint via SAA approach. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1653-1675. doi: 10.3934/jimo.2018116 [9] Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (2) : 485-491. doi: 10.3934/jimo.2012.8.485 [10] Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203 [11] Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331 [12] Jun Fan, Dao-Hong Xiang. Quantitative convergence analysis of kernel based large-margin unified machines. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4069-4083. doi: 10.3934/cpaa.2020180 [13] María Teresa V. Martínez-Palacios, Adrián Hernández-Del-Valle, Ambrosio Ortiz-Ramírez. On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach. Journal of Dynamics & Games, 2019, 6 (1) : 53-64. doi: 10.3934/jdg.2019004 [14] Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 317-331. doi: 10.3934/naco.2011.1.317 [15] Håkon Hoel, Gaukhar Shaimerdenova, Raúl Tempone. Multilevel Ensemble Kalman Filtering based on a sample average of independent EnKF estimators. Foundations of Data Science, 2020, 2 (4) : 351-390. doi: 10.3934/fods.2020017 [16] Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47 [17] George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations & Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557 [18] T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675 [19] Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial & Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673 [20] Jianlin Jiang, Shun Zhang, Su Zhang, Jie Wen. A variational inequality approach for constrained multifacility Weber problem under gauge. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1085-1104. doi: 10.3934/jimo.2017091

2020 Impact Factor: 1.801

Article outline