Scalarization of Mixed Vector Variational Inequalities and Error Bounds of Gap Functions

LIU Dan-yang; JIANG Ya

doi:10.21656/1000-0887.370292

Volume 38 Issue 6

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2017 > 38(6): 715-726

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

Citation:

PDF( 399 KB)

Scalarization of Mixed Vector Variational Inequalities and Error Bounds of Gap Functions

doi: 10.21656/1000-0887.370292

College of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637002, P.R.China

Funds: The National Natural Science Foundation of China（11371015）

Received Date: 2016-09-23
Rev Recd Date: 2017-05-10
Publish Date: 2017-06-15

Abstract

Abstract

The Konnov scalarization method for variational inequality problems was used to further generalize the classical strongly variational inequalities (SVIs) and the classical weakly variational inequalities (WVIs). The strongly generalized mixed vector variational inequalities (SGMVVIs) and the weakly generalized mixed vector variational inequalities (WGMVVIs) were studied based on setvalued mappings in view of their gap functions. Under proper conditions, the relationship between the gap function of the strongly generalized mixed setvalued variational inequality (SGMVI) and that of the SGMVVI, and the relationship between the gap functions of the WGMVVI and the SGMVI, were discussed. At last, the global error bounds of the gap functions were obtained.
- generalized mixed vector variational inequality,
- set-valued mapping,
- scalarization,
- gap function,
- error bound

FullText(HTML)

References(25)

References

[1]	Konnov I V. A scalarization approach for vector variational inequalities with applications[J]. Journal of Global Optimization,2005,32(4): 517-527.
[2]	Soleimani-Damaneh M. The gap function for optimization problems in Banach spaces[J].Nonlinear Analysis: Theory, Methods & Applications,2007,69(2): 716-723.
[3]	FAN Jiang-hua, WANG Xiao-guo. Gap functions and global error bounds for set-valued variational inequalities[J]. Journal of Computational and Applied Mathematics,2010,233(11): 2956-2965.
[4]	Solodov M V. Merit functions and error bounds for generalized variational inequalities[J]. Journal of Mathematic Analysis and Applications,2003,287(2): 405-414.
[5]	LI Ming-hua. Error bounds of regularized gap functions for weak vector variational inequality problems[J]. Journal of Inequalities and Applications,2014,2014: 331.
[6]	Mastroeni G. Gap functions for equilibrium problems[J]. Journal of Global Optimization, 2003,27(4): 411-426.
[7]	Konnov I V. On the generalized vector variational inequality problem[J]. Journal of Mathematic Analysis and Applications,1997,206(1): 42-58.
[8]	Giannessi F. Theorems of the alternative, quadratic programs, and complementarity problems[M]//Cottle R W, Giannessi F, Lion J L, ed. Variational Inequality and Complementarity Problems.Chichester, England: John Wiley and Sons, 1980: 151-186.
[9]	ZHANG Wen-yan, CHEN Jia-wei, XU Shu, et al. Scalar gap functions and error bounds for generalized mixed vector equilibrium problems with applications[J]. Fixed Point Theory and Applications,2015,2015: 169.
[10]	Chen G Y. Existence of solutions for a vector variational inequality: an extension of Hartmann-Stampacchia theorem[J]. Journal of Optimization Theory and Application,1992,74(3): 445-456.
[11]	Yang X Q, Yao J C. Gap functions and existence of solutions to set-valued vector variational inequalities[J]. Journal of Optimization Theory and Application,2002,115(2): 407-417.
[12]	CHEN Guang-ya, YANG Xiao-qi. Vector complementarity problem and its equivalences with weak minimal element in ordered spaces[J]. Journal of Mathematic Analysis and Applications,1990,153(1): 136-158.
[13]	CHEN Guang-ya, CHENG Ging-min. Vector variational inequality and vector optimization problem[M]// Lecture Notes in Economics and Mathematical Systems . Springer-Verlag, Heidelberg, 1987,258: 408-416.
[14]	Chen G Y, Craven B D. A vector variational inequality and optimization over an efficient set[J]. Zeitschrift fur Operation Research,1990,34(1): 1-12.
[15]	Yang X Q. Vector variational inequality and its duality[J]. Nonlinear Analysis: Theory, Methods & Applications,1993,21(11): 869-877.
[16]	Yu S J, Yao J C. On vector variational inequalities[J]. Journal of Optimization Theory and Applications,1996,89(3): 749-769.
[17]	Li J, Mastroeni G. Vector variational inequalities involving set-valued mappings via scalarization with applications to error bounds for gap functions[J]. Journal of Optimization Theory and Application,2010,145(2): 355-372.
[18]	Li J, Huang N J, Yang X Q. Weak sharp minima for set-valued vector variational inequalitieswith an application[J]. European Journal of Operational Research,2010,205(2): 262-272.
[19]	TANG Guo-ji, HUANG Nan-jing. Gap function and global error bounds for set-valued mixed variational inequalities[J]. Taiwanese Journal of Mathematics,2013,17(4): 1267-1286.
[20]	Zalinescu C. Convex Analysis in General Vector Spaces [M]. World Scientific Publishing Co Inc, 2002.
[21]	Rockafellar R T, Wets R J-B. Variational Analysis [M]. Springer-Verlag Berlin Heidelberg, 1998.
[22]	Kneser H. Sur un theoreme fondamental de la theorie des jeux C R[J]. Academy Science Paris,1952,234: 2418-2420.
[23]	LI Guo-yin, TANG Chun-ming, WEI Zeng-xin. Error bound results for generalized D-gap functions of nonsmooth variational inequality problems[J]. Journal of Computational and Applied Mathematics,2010,233(11): 2795-2806.
[24]	Chen G Y, Goh C J, Yang X Q. On gap functions for vector variational inequalities[M]//Giannessi F, ed. Vector Variational Inequalities and Vector Equilibria. Nonconvex Optimization and Its Applications,Vol38. Springer US, 2000: 55-72.
[25]	夏道行, 吴卓人, 严绍宗, 等. 实变函数与泛函分析[M]. 北京: 高等教育出版社, 1978.(XIA Dao-xin, WU Zhuo-ren, YAN Shao-zong, et al. Real Analysis and Functional Analysis [M]. Beijing: Higher Education Press, 1978.(in Chinese))