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Existence and convergence for stochastic differential variational inequalities

Year 2023, Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications, 1461 - 1479, 03.11.2023
https://doi.org/10.15672/hujms.1141495

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.

References

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  • [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] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, NorthHolland Publishing, Amsterdam, 1973.
  • [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] 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] 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] 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] 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] F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Verlag, New York, 2003.
  • [10] A.F. Filippov, On certain questions in the theory of optimal control, SIAM J. Control Optim. 1 (1), 76-84, 1962.
  • [11] G. Gordon and R. Tibshirani, Karush-Kuhn-Tucker conditions, Optimization. 10 (725/36), 725, 2012.
  • [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] 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] L.A. Korf and R.J-B. Wets, Random lsc functions: an ergodic theorem, Math. Oper. Res. 26 (2), 421-445, 2001.
  • [15] S. Lang, Real and Functional Analysis, Springer, Berlin, 1993.
  • [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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] Z.H. Liu and S.D. Zeng, Differential variational inequalities in infinite banach spaces, Acta Math. Sci. 37 (1), 26-32, 2017.
  • [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] G.D. Maso, An Introduction to Γ-Convergence, Springer-Verlag, New York, 1993.
  • [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] A. Nagurney, Network Economics: A Variational Inequality Approach, Kluwer Aca- demic Publisher, Dordrecht, The Netherlands, Second and Revised Edition, 1999.
  • [32] J.S. Pang and D.E. Stewart, Differential variational inequalities, Math. Program. 113 (2), 345-424, 2008.
  • [33] M. Patriksson, Nonlinear Programming and Variational Inequalities: A Unified Ap- proach, Dordrecht: Kluwer Academic Publishers, 1999.
  • [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] Z. Peng, Z.H. Liu and X. Liu, Boundary hemivariational inequality problems with doubly nonlinear operators, Math. Ann. 356 (4), 1339-1358, 2013.
  • [36] L.R. Petzold and U.M. Ascher, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM Publications, Philadelphia, 1998.
  • [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] U. Shanbhag, Stochastic Variational Inequality Problems: Applications, Analysis and Algorithms, Tutorials in Operations Research, INFORMS 2013.
  • [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] A. Shapiro, D. Dentcheva and A. Ruszczynski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, 2009.
  • [41] K.R. Stromberg, Introduction to Classical Real Analysis, Wadsworth, Inc. Belmont, 1981.
  • [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] 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] 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.
Year 2023, Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications, 1461 - 1479, 03.11.2023
https://doi.org/10.15672/hujms.1141495

Abstract

References

  • [1] J.P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer, Berlin, 1984.
  • [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] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, NorthHolland Publishing, Amsterdam, 1973.
  • [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] 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] 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] 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] 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] F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Verlag, New York, 2003.
  • [10] A.F. Filippov, On certain questions in the theory of optimal control, SIAM J. Control Optim. 1 (1), 76-84, 1962.
  • [11] G. Gordon and R. Tibshirani, Karush-Kuhn-Tucker conditions, Optimization. 10 (725/36), 725, 2012.
  • [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] 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] L.A. Korf and R.J-B. Wets, Random lsc functions: an ergodic theorem, Math. Oper. Res. 26 (2), 421-445, 2001.
  • [15] S. Lang, Real and Functional Analysis, Springer, Berlin, 1993.
  • [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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] Z.H. Liu and S.D. Zeng, Differential variational inequalities in infinite banach spaces, Acta Math. Sci. 37 (1), 26-32, 2017.
  • [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] G.D. Maso, An Introduction to Γ-Convergence, Springer-Verlag, New York, 1993.
  • [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] A. Nagurney, Network Economics: A Variational Inequality Approach, Kluwer Aca- demic Publisher, Dordrecht, The Netherlands, Second and Revised Edition, 1999.
  • [32] J.S. Pang and D.E. Stewart, Differential variational inequalities, Math. Program. 113 (2), 345-424, 2008.
  • [33] M. Patriksson, Nonlinear Programming and Variational Inequalities: A Unified Ap- proach, Dordrecht: Kluwer Academic Publishers, 1999.
  • [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] Z. Peng, Z.H. Liu and X. Liu, Boundary hemivariational inequality problems with doubly nonlinear operators, Math. Ann. 356 (4), 1339-1358, 2013.
  • [36] L.R. Petzold and U.M. Ascher, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM Publications, Philadelphia, 1998.
  • [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] U. Shanbhag, Stochastic Variational Inequality Problems: Applications, Analysis and Algorithms, Tutorials in Operations Research, INFORMS 2013.
  • [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] A. Shapiro, D. Dentcheva and A. Ruszczynski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, 2009.
  • [41] K.R. Stromberg, Introduction to Classical Real Analysis, Wadsworth, Inc. Belmont, 1981.
  • [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] 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] 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.
There are 44 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Fei Guan 0000-0001-7582-8468

Van Thien Nguyen 0000-0002-9670-1359

Zijia Peng 0000-0003-1576-2828

Publication Date November 3, 2023
Published in Issue Year 2023 Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications

Cite

APA Guan, F., Nguyen, V. T., & Peng, Z. (2023). Existence and convergence for stochastic differential variational inequalities. Hacettepe Journal of Mathematics and Statistics, 52(6), 1461-1479. https://doi.org/10.15672/hujms.1141495
AMA Guan F, Nguyen VT, Peng Z. Existence and convergence for stochastic differential variational inequalities. Hacettepe Journal of Mathematics and Statistics. November 2023;52(6):1461-1479. doi:10.15672/hujms.1141495
Chicago Guan, Fei, Van Thien Nguyen, and Zijia Peng. “Existence and Convergence for Stochastic Differential Variational Inequalities”. Hacettepe Journal of Mathematics and Statistics 52, no. 6 (November 2023): 1461-79. https://doi.org/10.15672/hujms.1141495.
EndNote Guan F, Nguyen VT, Peng Z (November 1, 2023) Existence and convergence for stochastic differential variational inequalities. Hacettepe Journal of Mathematics and Statistics 52 6 1461–1479.
IEEE F. Guan, V. T. Nguyen, and Z. Peng, “Existence and convergence for stochastic differential variational inequalities”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 6, pp. 1461–1479, 2023, doi: 10.15672/hujms.1141495.
ISNAD Guan, Fei et al. “Existence and Convergence for Stochastic Differential Variational Inequalities”. Hacettepe Journal of Mathematics and Statistics 52/6 (November 2023), 1461-1479. https://doi.org/10.15672/hujms.1141495.
JAMA Guan F, Nguyen VT, Peng Z. Existence and convergence for stochastic differential variational inequalities. Hacettepe Journal of Mathematics and Statistics. 2023;52:1461–1479.
MLA Guan, Fei et al. “Existence and Convergence for Stochastic Differential Variational Inequalities”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 6, 2023, pp. 1461-79, doi:10.15672/hujms.1141495.
Vancouver Guan F, Nguyen VT, Peng Z. Existence and convergence for stochastic differential variational inequalities. Hacettepe Journal of Mathematics and Statistics. 2023;52(6):1461-79.