Error bounds for a class of history-dependent variational inequalities controlled by $\mathcal{D}$-gap~functions

Year 2023, Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications, 1550 - 1566, 03.11.2023

Boling Chen Vo Minh Tam

https://doi.org/10.15672/hujms.1183739

Cited By: 2

https://izlik.org/JA37JP35AU

Abstract

In the present paper, we are concerned with investigating error bounds for history-dependent variational inequalities controlled by the difference gap (for brevity, $\mathcal{D}$-gap) functions. First, we recall a class of elliptic variational inequalities involving the history-dependent operators (for brevity, HDVI). Then, we introduce a new concept of gap functions to the HDVI and propose the regularized gap function for the HDVI via the optimality condition for the concerning minimization problem. Consequently, the $\mathcal{D}$-gap function for the HDVI depends on these regularized gap functions is established. Finally, error bounds for the HDVI controlled by the regularized gap function and the $\mathcal{D}$-gap function are derived under suitable conditions.

Keywords

History-dependent operator , Gap function , Error bound , Variational inequality

References

• [1] G. Bigi and M. Passacantando, D-gap functions and descent techniques for solving equilibrium problems, J. Global Optim. 62 (1), 183–203, 2015.
• [2] J.X. Cen, A.A. Khan, D. Motreanu and S.D. Zeng, Inverse problems for generalized quasi-variational inequalities with application to elliptic mixed boundary value systems, Inverse Problems 38, 065006, 2022.
• [3] J.X. Cen, V.T. Nguyen and S.D. Zeng, Gap functions and global error bounds for history-dependent variational-hemivariational inequalities, J. Nonlinear Var. Anal. 6, 461–481, 2022.
• [4] C. Charitha, A note on D-gap functions for equilibrium problems, Optimization, 62 (2), 211–226, 2013.
• [5] Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.
• [6] Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.
• [7] M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Program. 53 (4), 99–110, 1992.
• [8] J. Haslinger, M. Miettinen and P.D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications. Kluwer Academic Publishers, Boston, 1999.
• [9] N.V. Hung, S. Migórski, V.M. Tam and S. Zeng, Gap functions and error bounds for variational-hemivariational inequalities, Acta. Appl. Math. 169, 691–709, 2020.
• [10] N.V. Hung and V.M. Tam, Error bound analysis of the D-gap functions for a class of elliptic variational inequalities with applications to frictional contact mechanics, Z. Angew. Math. Phys. 72, 173, 2021.
• [11] N.V. Hung, V.M. Tam and B. Dumitru, Regularized gap functions and error bounds for split mixed vector quasivariational inequality problems, Math. Methods Appl. Sci. 43, 4614–4626, 2020.
• [12] N.V. Hung, V.M. Tam and Y. Zhou, A new class of strong mixed vector GQVIP- generalized quasi-variational inequality problems in fuzzy environment with regularized gap functions based error bounds, J Comput Appl Math. 381, 113055, 2021.
• [13] N.V. Hung, X. Qin, V.M. Tam and J.C. Yao, Difference gap functions and global error bounds for random mixed equilibrium problems, Filomat 34, 2739–2761, 2020.
• [14] I.V. Konnov and O.V. Pinyagina, D-gap functions for a class of equilibrium problems in Banach spaces, Comput. Methods Appl. Math. 3 (2), 274–286, 2003.
• [15] E.S. Levitin and B.T. Polyak, Constrained minimization methods, Comput. Math. Math. Phys. 6, 1–50, 1996.
• [16] G. Li and K.F. Ng, Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems, SIAM J. Optim. 20 (2), 667–690, 2009.
• [17] G. Li, C. Tang and Z. Wei, Error bound results for generalized D-gap functions of nonsmooth variational inequality problems, J. Comput. Appl. Math. 233 (11), 2795– 2806, 2010.
• [18] Z.H. Liu, D. Motreanu and S.D. Zeng, Generalized penalty and regularization method for differential variational-hemivariational inequalities, SIAM J. Optim. 31, 1158– 1183, 2021.
• [19] Z.Q. Luo and P. Tseng, Error bounds and convergence analysis of feasible descent methods: A general approach, Ann. Oper. Res. 46, 157–178, 1993.
• [20] S. Migórski, Y. Bai and S.D. Zeng, A new class of history-dependent quasi variational-hemivariational inequalities with constraints, Commun. Nonlinear Sci. Numer. Simul. 114, 106686, 2022.
• [21] S. Migórski and S.D. Zeng, A class of differential hemivariational inequalities in Ba- nach spaces, J. Glob. Optim. 72, 761–779, 2018.
• [22] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, in: Advances in Mechanics and Mathematics 26, Springer, New York, 2013.
• [23] S. Migórski, A. Ochal and M. Sofonea, History-dependent variational-hemivariational inequalities in contact mechanics, Nonlinear Anal. Real World Appl. 22, 604–618, 2015.
• [24] J.M. Peng, Equivalence of variational inequality problems to unconstrained minimization, Math. Program. 78 (3), 347–355, 1997.
• [25] J.M. Peng and M. Fukushima, A hybrid Newton method for solving the variational inequality problem via the D-gap function, Math. Program. 86 (2), 367–386, 1999.
• [26] M.V. Solodov and P. Tseng, Some methods based on the D-gap function for solving monotone variational inequalities, Comput. Optim. Appl. 17 (2–3), 255–277, 2000.
• [27] M. Sofonea, S. Migórski and W. Han, A penalty method for history-dependent variational-hemivariational inequalities, Comput. Math. Appl. 75 (7), 2561–2573, 2018.
• [28] M. Sofonea and F. Pătrulescu, Penalization of history-dependent variational inequal- ities, Eur. J. Appl. Math. 25 (2), 155–176, 2014.
• [29] M. Sofonea, W. Han and S. Migórski, Numerical analysis of history-dependent variational-hemivariational inequalities with applications to contact problems, Eur. J. Appl. Math. 26 (4), 427–452, 2015.
• [30] M. Sofonea and A. Matei, History-dependent quasi-variational inequalities arising in contact mechanics, Eur. J. Appl. Math. 22, (5), 471–491, 2011.
• [31] M. Sofonea and Y.-B. Xiao, Fully history-dependent quasivariational inequalities in contact mechanics, Appl. Anal. 95 (11), 2464–2484, 2016.
• [32] V.M. Tam, Upper-bound error estimates for double phase obstacle problems with Clarke’s subdifferential, Numer. Funct. Anal. Optim. 43 (4), 463–485, 2022.
• [33] P. Tseng, On linear convergence of iterative methods for the variational inequality, J. Comput. Appl. Math. 60, 237–252, 1995.
• [34] F.P. Vasil’yev, Methods of Solution of Extremal Problems, Nauka, Moscow, 1981.
• [35] J.H. Wu, M. Florian and P. Marcotte, A general descent framework for the monotone variational inequality problem, Math. Program. 61, 281–300, 1993.
• [36] N. Yamashita and M. Fukushima, Equivalent unconstrained minimization and global error bounds for variational inequality problems, SIAM J. Control Optim. 35, 273– 284, 1997.
• [37] S.D. Zeng, Y.R. Bai, L. Gasiński and P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calc. Var. PDEs 59(5), 1–18, 2020.
• [38] 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, 2829–2862, 2021.
• [39] S.D. Zeng, N.S. Papageorgiou and V.D. Rădulescu, Nonsmooth dynamical systems: From the existence of solutions to optimal and feedback control, Bull. Sci. Math. 176, 103131, 2022.
• [40] S.D. Zeng, V.D. Rădulescu and P. Winkert, Double phase implicit obstacle problems with convection and multivalued mixed boundary value conditions, SIAM J. Math. Anal. 54, 1898–1926, 2022.
• [41] S.D. Zeng and E. Vilches, Well-posedness of history/state-dependent implicit sweeping processes, J. Optim. Theory Appl. 186, 960–984, 2020.