广义混合拟变分不等式的间隙函数及其误差界

doi:10.15960/j.cnki.issn.1007-6093.2025.02.003

运筹学学报（中英文） ›› 2025, Vol. 29 ›› Issue (2): 44-57.doi: 10.15960/j.cnki.issn.1007-6093.2025.02.003

广义混合拟变分不等式的间隙函数及其误差界

陈巧¹, 林惠玲¹^,*()

1. 福建师范大学数学与统计学院, 福建福州 350117

收稿日期:2022-03-15 出版日期:2025-06-15 发布日期:2025-06-12
通讯作者: 林惠玲 E-mail:hllin@fjnu.edu.cn
基金资助:
福建省自然科学基金(2021J01654)

Gap functions and error bounds for generalized mixed quasi-variational inequalities

Qiao CHEN¹, Huiling LIN¹^,*()

1. School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, Fujian, China

Received:2022-03-15 Online:2025-06-15 Published:2025-06-12
Contact: Huiling LIN E-mail:hllin@fjnu.edu.cn

摘要/Abstract

摘要：

广义混合拟变分不等式(GMQVI) 是变分不等式的推广。变分不等式及其推广模型被广泛地应用在经济学、交通、最优控制等领域。基于广义投影算子的相关性质和GMQVI解的特征, 在一定的单调性和紧性假设下, 本文在自反Banach空间中给出了问题(GMQVI) 的正则化间隙函数, 并由此构造了D-间隙函数。同时利用这些间隙函数, 得到了基于正则化间隙函数的局部误差界, 和利用D-间隙函数及正则项刻画的全局误差界。

关键词: 广义混合拟变分不等式, 间隙函数, 误差界

Abstract:

A generalized mixed quasi-variational inequality (GMQVI) is a generalization of the variational inequality. Variational inequalities and their generalization are widely used, for instance, economics, transportation and optimal control. Based on the intrinsic properties of the generalized projection operator, under certain assumptions of monotonicity and compactness, this paper gives the regularized gap function for generalized mixed quasi-variational inequality problems in reflexive Banach spaces, which in turn induces the D-gap function for GMQVI by the difference of two regularized gap functions. And using these gap functions, the local error bound described by the regularized gap function and the global error bound characterized by the D-gap function and the regularization term are obtained.

Key words: generalized mixed quasi-variational inequalities, gap functions, error bounds

中图分类号:

O221.2

陈巧, 林惠玲. 广义混合拟变分不等式的间隙函数及其误差界[J]. 运筹学学报（中英文）, 2025, 29(2): 44-57.

Qiao CHEN, Huiling LIN. Gap functions and error bounds for generalized mixed quasi-variational inequalities[J]. Operations Research Transactions, 2025, 29(2): 44-57.

参考文献 36

1	Aussel D , Cao Van K , Salas D . Quasi-variational inequality problems over product sets with quasi-monotone operators[J]. SIAM Journal on Optimization, 2019, 29 (2): 1558- 1577. doi: 10.1137/18M1191270
2	李艳, 夏福全. Banach空间中广义混合变分不等式解的迭代算法[J]. 四川师范大学学报(自然科学版), 2011, 34 (1): 13- 19. doi: 10.3969/j.issn.1001-8395.2011.01.003
3	罗雪萍. 自反巴拿赫空间中混合变分不等式的稳定性与Tikhonov正则化[J]. 西南民族大学学报(自然科学版), 2017, 43 (6): 612- 617.
4	Fakhar M , Zafarani J . On generalized variational inequalities[J]. Journal of Global Optimization, 2009, 43 (4): 503- 511. doi: 10.1007/s10898-008-9346-2
5	翁姗. 混合拟变分不等式及其应用[D]. 桂林: 广西师范大学, 2017.
6	Robinson S M. Generalized equations and their solutions, part Ⅰ: Basic theory[M]//Huard P (ed.). Point-to-Set Maps and Mathematical Programming, Mathematical Programming Studies, 1979, 10: 128-141.
7	邵长英, 黄力人. 变分不等式问题与无约束最优化问题[J]. 海南师范学院学报(自然科学版), 2002, 15 (2): 1- 8.
8	袁媛媛, 张文伟, 屈彪. 求解拟变分不等式问题的一种外梯度算法[J]. 应用数学进展, 2015, 4 (1): 70- 75.
9	Kanzow C , Steck D . Quasi-variational inequalities in Banach spaces: Theory and augmented Lagrangian methods[J]. SIAM Journal on Optimization, 2019, 29 (4): 3174- 3200. doi: 10.1137/18M1230475
10	侯丽娜, 孙海琳. 交通网络下的多厂商两阶段随机非合作博弈问题——基于随机变分不等式[J]. 运筹学学报, 2019, 23 (3): 91- 108. doi: 10.15960/j.cnki.issn.1007-6093.2019.03.007
11	Alphonse A , Hintermüller M , Rautenberg C N . Stability of the solution set of quasi-variational inequalities and optimal control[J]. SIAM Journal on Control and Optimization, 2020, 58 (6): 3508- 3532. doi: 10.1137/19M1250327
12	Facchinei F , Pang J S , Scutari G , et al. Ⅵ-constrained hemivariational inequalities: Distributed algorithms and power control in ad-hoc networks[J]. Mathematical Programming, 2014, 145, 59- 96. doi: 10.1007/s10107-013-0640-5
13	Wang Z B , Xiao Y B , Chen Z Y . Degree theory for generalized mixed quasi-variational inequalities and its applications[J]. Journal of Optimization Theory and Applications, 2020, 187 (1): 43- 64. doi: 10.1007/s10957-020-01748-0
14	赵渊嫣. 广义变分不等式解的精炼及应用[D]. 贵阳: 贵州大学, 2016.
15	李小焕, 何洪津, 韩德仁. 一种改进的自适应投影法解广义纳什均衡问题[J]. 南京师范大学学报(自然科学版), 2011, 34 (2): 10- 14.
16	Kubota K , Fukushima K . Gap function approach to the generalized Nash equilibrium problem[J]. Journal of Optimization Theory and Applications, 2010, 144 (3): 511- 531. doi: 10.1007/s10957-009-9614-4
17	Gupta R , Mehra A . Gap functions and error bounds for quasi variational inequalities[J]. Journal of Global Optimization, 2012, 53 (4): 737- 748. doi: 10.1007/s10898-011-9733-y
18	Khan S A , Chen J W . Gap function and global error bounds for generalized mixed quasi variational inequalities[J]. Applied Mathematics and Computation, 2015, 260, 71- 81. doi: 10.1016/j.amc.2015.03.056
19	Aussel D , Correa R , Marechal M . Gap functions for quasivariational inequalities and generalized nash equilibrium problems[J]. Journal of Optimization Theory and Applications, 2011, 151 (3): 474- 488. doi: 10.1007/s10957-011-9898-z
20	童裕孙. 泛函分析教程[M]. 上海: 复旦大学出版社, 2007.
21	李曦, 黄南京, 邹云志. 广义f-投影算子的稳定性及其应用[J]. 数学学报, 2011, 54 (5): 811- 822.
22	唐国吉, 黄南京. 非Lipschitz集值混合变分不等式的一个投影次梯度方法[J]. 应用数学和力学, 2011, 32 (10): 1254- 1264. doi: 10.3879/j.issn.1000-0887.2011.10.011
23	方长杰, 郑继明, 吴慧莲. Banach空间中一类广义集值非线性混合似变分不等式解的存在性与算法[J]. 四川师范大学学报(自然科学版), 2007, 30 (1): 40- 44.
24	吴绍平. 泛函分析及其应用[M]. 浙江: 浙江大学出版社, 1990: 12.
25	王元明. 非线性偏微分方程: 上册[M]. 南京: 东南大学出版社, 1992: 140.
26	李志龙, 王秀梅. 一个变分定理的随机化[J]. 苏州科技学院学报, 2003 (2): 17- 20.
27	韩小琴, 冯世强. Banach空间中广义混合变分不等式的新逼近算法[J]. 乐山师范学院学报, 2015, 30 (8): 1- 4.
28	张冬杨, 关伟波. Banach空间中广义f-投影算子的连续性及其应用[J]. 哈尔滨师范大学自然科学学报, 2013, 29 (1): 5-7, 11.
29	Royden H L . Real Analysis[M]. New York: Macmillan Publishing Company, 1988: 159- 161.
30	Li X , Li X S , Huang N J . A generalized f-projection algorithm for inverse mixed variational inequalities[J]. Optimization Letters, 2014, 8 (3): 1063- 1076.
31	Taji K . On gap functions for quasi-variational inequalities[J]. Abstract and Applied Analysis, 2008 (1): 1- 7.
32	Anh L Q , Hung N V , Tam V M . Regularized gap functions and error bounds for generalized mixed strong vector quasiequilibrium problems[J]. Computational and Applied Mathematics, 2018, 34 (5): 5935- 5950.
33	林清英. 集值拟变分不等式的间隙函数及误差边界[D]. 厦门: 集美大学, 2014.
34	Kubotak K , Fukushima M . Gap function approach to the generalized Nash equilibrium problem[J]. Journal of Optimization Theory and Applications, 2010, 144 (3): 511- 531.
35	Solodov M V . Merit functions and error bounds for generalized variational inequalities[J]. Journal of Mathematical Analysis and Applications, 2003, 287 (2): 405- 414.
36	Hiriart J B, Lemaréchal C. Convex Analysis and Minimization Algorithms Ⅰ Volume 305[M]. Berlin: Springer, 1993: 267.

广义混合拟变分不等式的间隙函数及其误差界

Gap functions and error bounds for generalized mixed quasi-variational inequalities

RichHTML

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献 36

相关文章 1

编辑推荐

Metrics

本文评价