Hadamard Well-posedness for a Family of Mixed Variational Inequalities and Inclusion Problems‎

Subject Areas : International Journal of Industrial Mathematics

الهام  خاکراه ¹ (گروه ریاضی، دانشگاه بین المللی امام خمینی قزوین، قزوین، ایران.)
عبدالرحمان  رازانی ² (گروه ریاضی، دانشگاه بین المللی امام خمینی قزوین، قزوین، ایران.)
مرتضی  اویسیها ³ (گروه ریاضی، دانشگاه بین المللی امام خمینی قزوین، قزوین، ایران.)

Received: 2019-07-17 Accepted : 2020-10-14 Published : 2021-01-01

Keywords: Monotonicity, Approximating sequence, Mixed variational inequality, Parametric well-posedness,

Abstract :

In this paper, the concepts of well-posednesses and Hadamard well-posedness for a family of mixed variational inequalities are studied. Also, some metric characterizations of them are presented and some relations between well-posedness and Hadamard well-posedness of a family of mixed variational inequalities is studied. Finally, a relation between well-posedness for the family of mixed variational inequalities and well-posedness for a family of inclusion problems is discussed.

References:

[1] H. Brezis, Operateurs maximaux monotone et semigroups de contractions dans les espaces de Hilbert, North-Holland, Amsterdam (1973).
[2] L. C. Ceng, N. Hadjisavvas, S. Schaible, J. C Yao, Well-posedness for mixed quasivariational-like inequalitie, J. Optim. Theory Appl. 139 (2008) 109-125.
[3] C. Chen, S. Ma, J. Yang, A general inertial proximal point algorithm for mixed variational inequality problem, SIAM J. Optim. 25 (2015) 2120-2142.
[4] M. Darabi, J. Zafarani, M-well-posedness and B-well-posedness for vector quasiequilibrium problems, J. Nonlinear Convex Anal. 17 (2016) 1607-1625.
[5] Y.P. Fang and C.X. Deng, Stability of new implicit iteration procedures for a class of nonlinear set-valued mixed variational inequalities, ZAMM. Z. Angew. Math. Mech. 84 (2004) 53-59.
[6] Y. P. Fang, R. Hu, Parametric wellposedness for variational inequalities dened by  bifunctions, Comput. Math. Appl. 53 (2007) 1306-1316.
[7] Y. P. Fang, N. J. Huang, J. C. Yao, Well posedness of mixed variational inequalities, inclusion problems and xed point problems, J. Global Optim. 41 (2008) 117-133.
[8] R. Glowinski, J. L. Lions, R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam (1981).
[9] R. Hu, Y. P. Fang, Levitin Polyak wellposedness by perturbations of inverse variational inequalities, Optim. Letters. 7 (2013) 343-359.
[10] X. X. Huang, X. Q. Yang, Generalized Levitin{Polyak well-posedness in constrained optimization, SIAM J. Optim. 17 (2006) 243-258.
[11] E. Khakrah, A. Razani, R. Mirzaei, M. Oveisiha, Some metric characterization of wellposedness for hemivariational-like inequalities, Journal of Nonlinear Functional Analysis, Article ID 44 (2017) 1-12.
[12] E. Khakrah, A. Razani, M. Oveisiha, Pseudoconvex multiobjective continuous{time problems and vector variational inequalities, Int. J. Industrial Mathematics 9 (2017) No.
3, Article ID IJIM-00861, 8 pages.
[13] M. B. Lignola, Well-posedness and L-wellposedness for quasivariational inequalities, J. Optim. Theory Appl. 128 (2006) 119-138.
[14] S. J. Li, W. Y. Zhang, Hadamard well{posedness vector optimization, J. Global Optim. 46 (2010) 383-393.
[15] X. B. Li, and F. Q. Xia, Hadamard well{posedness of a general mixed variational in-
equality in Banach space, J. Global Optim.56 (2013) 1617-1629.
[16] M.B. Lignola, J. Morgan, Approximate solutions to variational inequalities and application, Le Matematiche. 49 (1994) 161-220.
[17] M. B. Lignola, J. Morgan, Well-posedness for optimization problems with constraints dened by variational inequalities having a unique solution, J. Global Optim. 16 (2000) 57-67.
[18] R. Lucchetti, F. Patrone, Hadamard and Tykhonov well-posedness of certain class of
convex functions, J. Math. Anal. Appl. 88 (1982) 204-215.
[19] R. Lucchetti, F. Patrone, A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities, Numer. Funct. Anal. Optim. 3 (1981) 461-476.
[20] J. Morgan, Approximations and wellposedness in multicriteria games, Ann. Oper. Res. 137 (2005) 257-268.
[21] M. Oveisiha, A. Razani, E. Khakrah, Generalized convexities for multiobjective continuous-time problems and vector variational inequalities, 9th International Iranian Operations Research Conference (IORC 2016), IORC, April 27-29, Shiraz, Iran, (2016).
[22] M. Oveisiha, A. Razani, E. Khakrah, An Equivalence Result for Well-Posedness of Hemivariational-like Inequalities, Third International Conference on Nonlinear Analysis and optimization, May 25-27, Esfahan, Iran, (2015).
[23] J. W. Peng, S. Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems, Optim. Letters 4 (2010) 501-512.
[24] A. Razani, Results in Fixed Point Theory, Andisheh Zarin publisher, Qazvin, (2010).
[25] A. Razani, E. Khakrah, M. Oveisiha, Pascoletti-Serani scalarization and vector optimization with a variable ordering structure, Journal of Statistics and Management Systems 21 (2018) 917-931.
[26] A. N. Tykhonov, On the stability of the functional optimization problem, USSR J. Comput. Math. Phys. 6 (1966) 631-634.
[27] Y. B. Xiao, N. J. Huang, M. M. Wong, Wellposedness of hemivariational inequalities and inclusion problems, Taiwanese J. Math. 15 (2011) 1261-1276.
[28] H. Yang, J. Yu, Unied approaches to well posedness with some applications, J. GlobalOptim. 31 (2005) 371-383.
[29] G. X. Z. Yuan, KKM Theory and Applications to Nonlinear Analysis, Marcel Dekker, New York, (1999).