Scalar gap functions and error bounds for generalized mixed vector equilibrium problems with applications

Fixexd point theory and applications

Shu Xu¹  Wen Dong²  Jiawei Chen²  Wenyan Zhang³ 

[1] Education Committee of Banan District, Chongqing, China;School of Mathematics and Statistics, Southwest University, Chongqing, China;The School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, China

关键词: error bound;    scalar gap function;    generalized mixed vector equilibrium;   

DOI  :  10.1186/s13663-015-0422-2

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

It is well known that equilibrium problems are very important mathematical models and are closely related with fixed point problems, variational inequalities, and Nash equilibrium problems. Gap functions and error bounds which play a vital role in algorithms design, are two much-addressed topics of vector equilibrium problems. This paper is devoted to studying the scalar-valued gap functions and error bounds for the generalized mixed vector equilibrium problem (GMVE). First, a scalar gap function for (GMVE) is proposed without any scalarization methods, and then error bounds of (GMVE) are established in terms of the gap function. As applications, error bounds for generalized vector variational inequalities and vector variational inequalities are derived, respectively. The main results obtained are new and improve corresponding results of Charitha and Dutta (Pac. J. Optim. 6:497-510, 2010) and Sun and Chai (Optim. Lett. 8:1663-1673, 2014).

【 授权许可】

CC BY