Existence and convergence for stochastic differential variational inequalities

Abstract

In this paper, we consider a class of stochastic differential variational inequalities (for short, SDVIs) consisting of an ordinary differential equation and a stochastic variational inequality. The existence of solutions to SDVIs is established under the assumption that the leading operator in the stochastic variational inequality is $P$-function and $P_{0}$-function, respectively. Then, by using the sample average approximation and time stepping methods, two approximated problems corresponding to SDVIs are introduced and convergence results are obtained.

Keywords

• Stochastic differential variational inequality
• P-function
• convergence
• P_0-function

References

1. [1] J.P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer, Berlin, 1984.
2. [2] K.E. Brenan, S.L. Campbell and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential Algebraic Equations, SIAM Publications Classics in Applied Mathematics, Philadelphia, 1996.
3. [3] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, NorthHolland Publishing, Amsterdam, 1973.
4. [4] X.J. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Math. Oper. Res. 30 (4), 1022-1038, 2005.
5. [5] X.J. Chen, H.L. Sun and H.F. Xu, Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems, Math. Program. 177 (1), 255-289, 2019.
6. [6] X.J. Chen and Z.Y. Wang, Convergence of regularized time-stepping methods for differential variational inequalities, SIAM J. Optim. 23 (3), 1647-1671, 2013.
7. [7] X.J. Chen and Z.Y. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program. 146 (1), 379-408, 2014.
8. [8] X.J. Chen, R.J.-B. Wets and Y.F. Zhang, Stochastic variational inequalities: residual minimization smoothing sample average approximations, SIAM J. Optim. 22 (2), 649-673, 2012.

9. [9] F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Verlag, New York, 2003.
10. [10] A.F. Filippov, On certain questions in the theory of optimal control, SIAM J. Control Optim. 1 (1), 76-84, 1962.
11. [11] G. Gordon and R. Tibshirani, Karush-Kuhn-Tucker conditions, Optimization. 10 (725/36), 725, 2012.
12. [12] G. Haeser and M.L. Schuverdt, On approximate KKT condition and its extension to continuous variational inequalities, J. Optim. Theory Appl. 149 (3), 528-539, 2011.
13. [13] Y.R. Jiang, Q.Q. Song and Q.F. Zhang, Uniqueness and Hyers-Ulam stability of random differential variational inequalities with nonlocal boundary conditions, J. Optim. Theory Appl. 189 (2), 646-665, 2021.
14. [14] L.A. Korf and R.J-B. Wets, Random lsc functions: an ergodic theorem, Math. Oper. Res. 26 (2), 421-445, 2001.
15. [15] S. Lang, Real and Functional Analysis, Springer, Berlin, 1993.
16. [16] X.S. Li and N.J. Huang, A class of impulsive differential variational inequalities in finite dimensional spaces, J. Frankl. Inst. 353, 3151-3175, 2016.
17. [17] X.S. Li, N.J. Huang and D. O’Regan, Differential mixed variational inequalities in finite dimensional spaces, Nonlinear Anal. Theory Meth. Appl. 72 (9-10), 3875-3886, 2010.
18. [18] L.G. Li and W.H. Zhang, Study on indefinite stochastic linear quadratic optimal control with inequality constraint, J. Appl. Math. Comput. 2013, 4999-5004, 2013.
19. [19] X. Liu and Z.H. Liu Existence results for a class of second order evolution inclusions and its corresponding first order evolution inclusions, Israel J. Math. 194 (2), 723-743, 2013.
20. [20] Y.J. Liu, Z.H. Liu and D. Motreanu, Differential inclusion problems with convolution and discontinuous nonlinearities, Evolution Equations and Control Theory 9 (4), 1057-1071, 2020.
21. [21] Y.J. Liu, Z.H. Liu and C.-F. Wen, Existence of solutions for space-fractional parabolic hemivariational inequalities, Discrete Continuous Dynamical Systems - B 24 (3), 1297-1307, 2019.
22. [22] Y.J. Liu, Z.H. Liu, C.-F. Wen, J.-C. Yao and S.D. Zeng, Existence of solutions for a class of noncoercive variational-hemivariational inequalities arising in contact problems, Appl. Math. Optim. 84 (2), 2037-2059, 2021.
23. [23] Y.J. Liu, S. Migórski, V.T. Nguyen and S.D. Zeng, Existence and convergence results for an elastic frictional contact problem with nonmonotone subdifferential boundary conditions, Acta Math. Sci. 41 (4), 1151-1168, 2021.
24. [24] Z.H. Liu, S. Migórski and S.D. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in banach spaces, J. Differ. Equ. 263 (7), 3989- 4006, 2017.
25. [25] Z.H. Liu, D. Motreanu and S.D. Zeng, Generalized penalty and regularization method for differential variational-hemivariational inequalities, SIAM J. Optim. 31 (2), 1158- 1183, 2021.
26. [26] Z.H. Liu and B. Zeng, Optimal control of generalized quasi-variational hemivariational inequalities and its applications, Appl. Math. Optim. 72 (2), 305-323, 2015.
27. [27] Z.H. Liu and S.D. Zeng, Differential variational inequalities in infinite banach spaces, Acta Math. Sci. 37 (1), 26-32, 2017.
28. [28] J.F. Luo, Z.X. Wang and Y. Zhao, Convergence of discrete approximation for differential linear stochastic complementarity systems, Numerical Algorithms 87 (1), 223-262, 2021.
29. [29] G.D. Maso, An Introduction to Γ-Convergence, Springer-Verlag, New York, 1993.
30. [30] D. Motreanu and Z. Peng, Doubly coupled systems of elliptic hemivariational inequalities: existence and location, Comput. Math. Appl. 77 (11), 3001-3009, 2019.
31. [31] A. Nagurney, Network Economics: A Variational Inequality Approach, Kluwer Aca- demic Publisher, Dordrecht, The Netherlands, Second and Revised Edition, 1999.
32. [32] J.S. Pang and D.E. Stewart, Differential variational inequalities, Math. Program. 113 (2), 345-424, 2008.
33. [33] M. Patriksson, Nonlinear Programming and Variational Inequalities: A Unified Ap- proach, Dordrecht: Kluwer Academic Publishers, 1999.
34. [34] Z. Peng and Z.H. Liu, Evolution hemivariational inequality problems with doubly non- linear operators, J. Global Optim. 51 (3), 413-427, 2011.
35. [35] Z. Peng, Z.H. Liu and X. Liu, Boundary hemivariational inequality problems with doubly nonlinear operators, Math. Ann. 356 (4), 1339-1358, 2013.
36. [36] L.R. Petzold and U.M. Ascher, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM Publications, Philadelphia, 1998.
37. [37] R.T. Rockafellar and J. Sun, Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging, Math. Program. 174 (1), 453-471, 2019.
38. [38] U. Shanbhag, Stochastic Variational Inequality Problems: Applications, Analysis and Algorithms, Tutorials in Operations Research, INFORMS 2013.
39. [39] U. Shanbhag, J.S. Pang and S. Sen, Inexact best response scheme for stochastic nash games: linear convergence and iteration complexity analysis, IEEE Conference on Decision and Control, 3591-3596, 2016.
40. [40] A. Shapiro, D. Dentcheva and A. Ruszczynski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, 2009.
41. [41] K.R. Stromberg, Introduction to Classical Real Analysis, Wadsworth, Inc. Belmont, 1981.
42. [42] S.D. Zeng, Y.R. Bai, L. Gasínski and P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calc. Var. Partial Differ- ential Equations 59 (4), 176, 2020.
43. [43] S.D. Zeng, S. Migórski and Z.H. Liu, Well-Posedness, optimal control, and sensitivity analysis for a class of differential variational-Hemivariational inequalities, SIAM J. Optim. 31 (5), 2829-2862, 2021.
44. [44] S.D. Zeng, S. Migórski and Z.H. Liu, Nonstationary incompressible Navier-Stokes system governed by a quasilinear reaction-diffusion equation, Sci. Sin. Math. 52 (3), 331-354, 2022.