## Taiwanese Journal of Mathematics

### GAP FUNCTIONS AND GLOBAL ERROR BOUNDS FOR SET-VALUED MIXED VARIATIONAL INEQUALITIES

#### Abstract

In this paper, we introduce some gap functions for set-valued mixed variational inequalities under suitable conditions. We further use these gap functions to study global error bounds for the solutions of set-valued mixed variational inequalities in Hilbert spaces. The results presented in this paper generalize and improve some corresponding known results in literatures.

#### Article information

Source
Taiwanese J. Math., Volume 17, Number 4 (2013), 1267-1286.

Dates
First available in Project Euclid: 10 July 2017

https://projecteuclid.org/euclid.twjm/1499706116

Digital Object Identifier
doi:10.11650/tjm.17.2013.2247

Mathematical Reviews number (MathSciNet)
MR3085510

Zentralblatt MATH identifier
1304.90207

#### Citation

Huang, Nan-jing; Tang, Guo-ji. GAP FUNCTIONS AND GLOBAL ERROR BOUNDS FOR SET-VALUED MIXED VARIATIONAL INEQUALITIES. Taiwanese J. Math. 17 (2013), no. 4, 1267--1286. doi:10.11650/tjm.17.2013.2247. https://projecteuclid.org/euclid.twjm/1499706116

#### References

• J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhauser, Boston, 1990.
• D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1989.
• A. Chinchuluun, A. Migdalas, P. M. Pardalos and L. Pitsoulis, Pareto Optimality, Game Theory and Equilibria, Springer, Berlin, 2008.
• G. Cohen, Nash equilibria: gradient and decomposition algorithms, Large Scale Syst., 12 (1987), 173-184.
• A. Daniilidis and N. Hadjisavvas, Coercivity conditions and variational inequalities, Math. Program., 86 (1999), 433-438.
• F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementary Problems I, II, Springer-Verlag, New York, 2003.
• J. H. Fan and X. G. Wang, Gap functions and global error bounds for set-valued variational inequalities, J. Comput. Appl. Math., 233 (2010), 2956-2965.
• M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Program., 53 (1992), 99-110.
• F. Giannessi, A. Maugeri and P. M. Pardalos, Equilibrium Problems and Variational Models, Kluwer, Boston, 2001.
• W. Han and B. Reddy, On the finite element method for mixed variational inequalities arising in elastoplasticity, SIAM J. Numer. Anal., 32 (1995), 1778-1807.
• Y. R. He, A new projection algorithm for mixed variational inequalities, Acta Math. Sci., 27A (2007), 215-220.
• N. J. Huang, J. Li and S. Y. Wu, Gap functions for a system of generalized vector quasi-equilibrium problems with set-valued mappings, J. Glob. Optim., 41 (2008), 401-415.
• N. J. Huang, J. Li and J. C. Yao, Gap functions and existence of solutions for a system of vector equilibrium problems, J. Optim. Theory Appl., 133 (2007), 201-212.
• A. Kaplan and R. Tichatschke, Proximal point methods and nonconvex optimization, J. Glob. Optim., 13 (1998), 389-406.
• T. Larsson and M. Patriksson, A class of gap functions for variational inequalities, Math. Program., 64 (1994), 53-79.
• J. Li and N. J. Huang, An extension of gap functions for a system of vector equilibrium problems with applications to optimization problems, J. Glob. Optim., 39 (2007), 247-260.
• J. Li and G. Mastroeni, Vector variational inequalities involving set-valued mappings via scalarization with applications to error bounds for gap functions, J. Optim. Theory Appl., 145 (2010), 355-372.
• G. Li and K. F. Ng, On generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems, SIAM J. Optim., 20 (2009), 667-690.
• K. F. Ng and L. L. Tan, Error bounds of regularized gap functions for nonsmooth variational inequality problems, Math. Program., 110 (2007), 405-429.
• S. Nguyen and C. Dupuis, An efficient method for computing traffic equilibria in networks with asymmetric transportation costs, Transportation Sci., 18 (1984), 185-202.
• M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.
• M. V. Solodov, Merit functions and error bounds for generalized variational inequalities, J. Math. Anal. Appl., 287 (2003), 405-414.
• M. V. Solodov and B. F. Svaiter, Error bounds for proximal point subproblems and associated inexact proximal point algorithms, Math. Program., 88 (2000), 371-389.
• L. L. Tan, Regularized gap functions for nonsmooth variational inequality problems, J. Math. Anal. Appl., 334 (2007), 1022-1038.
• J. H. Wu, M. Florian and P. Marcotte, A general descent framework for the monotone variational inequality problem, Math. Program., 61 (1993), 281-300.
• K. Q. Wu and N. J. Huang, The generalized $f$-projection operator and set-valued variational inequalities in Banach spaces, Nonlinear Anal. TMA, 71 (2009), 2481-2490.
• F. Q. Xia, N. J. Huang and Z. B. Liu, A projected subgradient method for solving generalized mixed variational inequalities, Oper. Res. Lett., 36 (2008), 637-642.
• N. Yamashita and M. Fukushima, Equivalent uncontrained minimization and global error bounds for variational inequality problems, SIAM J. Control Optim., 35 (1997), 273-284.
• H. Y. Yin, C. X. Xu and Z. X. Zhang, The F-complementarity problem and its equivalence with the least element problem, Acta Math. Sinica, 44 (2001), 679-686.
• R. Y. Zhong and N. J. Huang, Stability analysis for Minty mixed variational inequality in reflexive Banach spaces, J. Optim. Theory Appl., 147 (2010), 454-472.