Generalized Gap Functions and Error Bounds for Generalized Variational Inequalities

HU Yan-hong; SONG Wen

Volume 30 Issue 3

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 2009 > 30(3): 301-308

HU Yan-hong, SONG Wen. Generalized Gap Functions and Error Bounds for Generalized Variational Inequalities[J]. Applied Mathematics and Mechanics, 2009, 30(3): 301-308.

Citation:

HU Yan-hong, SONG Wen. Generalized Gap Functions and Error Bounds for Generalized Variational Inequalities[J]. Applied Mathematics and Mechanics, 2009, 30(3): 301-308.

Citation:

HU Yan-hong, SONG Wen. Generalized Gap Functions and Error Bounds for Generalized Variational Inequalities[J]. Applied Mathematics and Mechanics, 2009, 30(3): 301-308.

PDF( 329 KB)

Generalized Gap Functions and Error Bounds for Generalized Variational Inequalities

HU Yan-hong^1,2,
SONG Wen^1,2

1.
Department of Mathematics, Northeast Normal University, Changchun 130024, P. R. China;

Received Date: 2008-08-18
Rev Recd Date: 2008-12-15
Publish Date: 2009-03-15

Abstract

Abstract

Some classes of generalized gap functions for two kinds of generalized variational inequality problems are considered. Error bounds for the underlying variational inequalities by using the generalized gap functions under the condition that the involved mapping F is gstrongly monotone with respect to the solution were obtained. It is not necessary to suppcsethat Fis continuously differential nor of local Lipschitz. with respect to the solution were obtained. It is not necessary to suppose that Fis continuously differentiable local Lipschitz.
- variational inequality,
- gap functions,
- error bounds

FullText(HTML)

References(8)

References

[1]	Fukushima M. Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems[J].Mathematical Programming,1992,53(1):99-110. doi: 10.1007/BF01585696
[2]	Wu J H,Florian M,Marcotte P.A general descent framework for the monotone variational inequality problem[J].Mathematical Programming,1993,61(3):281-300. doi: 10.1007/BF01582152
[3]	Yamashita N,Taji K,Fukushima M.Unconstrained optimization reformulations of variational inequality problems[J].Journal of Optimization Theory and Applications,1997,92(3):439-456. doi: 10.1023/A:1022660704427
[4]	Huang L R,Ng K F. Equivalent optimization formulations and error bounds for variational inequality problem[J].Journal of Optimization Theory and Applications,2005,125(2):299-314. doi: 10.1007/s10957-004-1839-7
[5]	Tan L L. Regularized gap functions for nonsmooth variational inequality problems[J].Journal of Mathematical Analysis and Applications,2007,334(2):1022-1038. doi: 10.1016/j.jmaa.2007.01.025
[6]	Solodov M V.Merit functions and error bounds for generalized variational inequalities[J].Journal of Mathematical Analysis and Applications,2003,287(2):405-414. doi: 10.1016/S0022-247X(02)00554-1
[7]	Noor M A. Merit functions for general variational inequalities[J].Journal of Mathematical Analysis and Applications,2006,316(2):736-752. doi: 10.1016/j.jmaa.2005.05.011
[8]	Qu B, Wang C Y,Zhang J Z. Convergence and error bound of a method for solving variational inequality problems via the generalized D-gap function[J].Journal of Optimization Theory and Applications,2003,119(3):535-552. doi: 10.1023/B:JOTA.0000006688.13248.04

Relative Articles

[1]	WANG Xiaoting, LONG Xianjun, PENG Zaiyun. A Double Projection Algorithm for Solving Non-Monotone Variational Inequalities[J]. Applied Mathematics and Mechanics, 2022, 43(8): 927-934. doi: 10.21656/1000-0887.420414
[2]	YANG Jun. A Golden Ratio Algorithm for Solving Nonmonotone Variational Inequalities[J]. Applied Mathematics and Mechanics, 2021, 42(7): 764-770. doi: 10.21656/1000-0887.410359
[3]	LIU Dan-yang, JIANG Ya. Scalarization of Mixed Vector Variational Inequalities and Error Bounds of Gap Functions[J]. Applied Mathematics and Mechanics, 2017, 38(6): 715-726. doi: 10.21656/1000-0887.370292
[4]	CHEN Guang-hua, CHEN Guang-ming, DAI Zhi-hua. Modified Domain Decomposition Method for Hamilton-Jacobi-Bellman Equations[J]. Applied Mathematics and Mechanics, 2010, 31(12): 1496-1502. doi: 10.3879/j.issn.1000-0887.2010.12.010
[5]	ZHANG Shi-sheng. On the Generalized Mixed Equilibrium Problem in Banach Spaces[J]. Applied Mathematics and Mechanics, 2009, 30(9): 1033-1041. doi: 10.3879/j.issn.1000-0887.2009.09.004
[6]	HAO Yan. Some Results on Variational Inclusion Problems and Fixed Point Problems With Applications[J]. Applied Mathematics and Mechanics, 2009, 30(12): 1493-1500. doi: 10.3879/j.issn.1000-0887.2009.12.010
[7]	LIU Ying. Strong Convergence Theorems for Relatively Nonexpansive Mappings and Inverse-Strongly-Monotone Mappings in a Banach Space[J]. Applied Mathematics and Mechanics, 2009, 30(7): 865-872. doi: 10.3879/j.issn.1000-0887.2009.07.011
[8]	ZHANG Shi-sheng, RAO Ruo-feng, HUANG Jia-lin. Strong Convergence Theorem for a Generalized Equilibrium Problem and k-Strict Pseudocontraction in Hilbert Spaces[J]. Applied Mathematics and Mechanics, 2009, 30(6): 638-647. doi: 10.3879/j.issn.1000-0887.2009.06.002
[9]	DING Xie-ping, LIN Yen-cherng, YAO Jen-chih. Three-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities[J]. Applied Mathematics and Mechanics, 2007, 28(8): 921-928.
[10]	ZHENG Hong, LIU De-fu, LEE C. F., THAM L. G.. New Variational Inequality Formulation for Seepage Problems With Free Surfaces[J]. Applied Mathematics and Mechanics, 2005, 26(3): 363-371.
[11]	ZHANG Shi-sheng. Multi-Valued Quasi Variational Inclusions in Banach Spaces[J]. Applied Mathematics and Mechanics, 2004, 25(6): 572-580.
[12]	WANG Zheng-yu, SHEN Zu-he. Kantorovich Theorem for Variational Inequalities[J]. Applied Mathematics and Mechanics, 2004, 25(11): 1182-1188.
[13]	GUO Xing-ming, ZHOU Shi-xing. Optimal Control of Parabolic Variational Inequalities With Stat Constraint[J]. Applied Mathematics and Mechanics, 2003, 24(7): 669-674.
[14]	Gao Yang. Contact Problems and Dual Variational Inequality of 2-D Elastoplastic Beam Theory[J]. Applied Mathematics and Mechanics, 1996, 17(10): 895-908.
[15]	Zheng Tie-sheng, Li Li, Xu Qing-yu. An Iterative Method for the Discrete Problems of a Class of Elliptical Variational Inequalities[J]. Applied Mathematics and Mechanics, 1995, 16(4): 329-335.
[16]	Zhang Shi-sheng, Zhang Yin. Section Theorems, Coincidence Theorems and Intersection Theorems on H-Spaces with Applications[J]. Applied Mathematics and Mechanics, 1993, 14(9): 763-774.
[17]	Guo Xiao-ming, She Ying-he. The Variational Inequality Formulation for the Deformation Theory in Plasticity and Its Non-Iterative Solution[J]. Applied Mathematics and Mechanics, 1993, 14(12): 1105-1113.
[18]	Wang Xiang-dong, Liang Xi-ting. The Holder Continuity for the Gradient of Solutions to One-Sided Obstacle Problems[J]. Applied Mathematics and Mechanics, 1992, 13(9): 811-820.
[19]	Zhang Shi-sheng, Xiang Shu-wen. On the Existence and Uniqueness of Solutions for a Class of Variational Inequalities with Applications to the Signorini Problem in Mechanics[J]. Applied Mathematics and Mechanics, 1991, 12(5): 401-407.
[20]	Sha De-song, Sun Huan-chun, Xu Shou-ze. Formulation of Boundary Element-Linear Complementary Equation for the Ffictional Elastic Contact Problems[J]. Applied Mathematics and Mechanics, 1990, 11(12): 1035-1041.