您的位置:山东大学 -> 科技期刊社 -> 《山东大学学报(理学版)》

山东大学学报(理学版) ›› 2018, Vol. 53 ›› Issue (2): 38-45.doi: 10.6040/j.issn.1671-9352.0.2017.310

• • 上一篇    下一篇

求解单调变分不等式问题的一种修正的次梯度超梯度方法

杨延涛   

  1. 延安大学数学与计算机科学学院, 陕西 延安 716000
  • 收稿日期:2017-06-21 出版日期:2018-02-20 发布日期:2018-01-31
  • 作者简介:杨延涛(1982— ),男,硕士,讲师,研究方向为非线性泛函分析. E-mail:yadxyyt@163.com
  • 基金资助:
    国家自然科学基金资助项目(11471007);陕西省教育厅2018年科学研究计划项目

Modified subgradient extragradient method for solving monotone variational inequality problems

YANG Yan-tao   

  1. College of Mathematics and Computer Science, Yanan University, Yanan 716000, Shaanxi, China
  • Received:2017-06-21 Online:2018-02-20 Published:2018-01-31

摘要: 提出了一种修正的次梯度超梯度方法,用以寻找非扩张映像不动点集与单调变分不等式解集之公共元,证明了由该算法所生成的迭代序列弱收敛于某公共元。所得结果改进并推广了已有文献的相关结果。

关键词: 单调变分不等式, 公共元, 弱收敛, 修正的次梯度超梯度方法, 非扩张映像

Abstract: A modified subgradient extragradient method for solving variational inequality problems is proposed in Hilbert space. It is shown that the sequence generated by the proposed algorithm converges weakly to a common element in the intersection of fixed points sets for nonexpansive mappings and of solution sets for monotone variational inequalities. The results presented in this paper improve and generalize the known results.

Key words: common element, monotone variational inequalities, weak convergence, nonexpansive mappings, modified subgradient extragradient method

中图分类号: 

  • O178
[1] KORPELEVICH G M.An extragradient method for finding saddle points and other problems[J]. Ekonomikai Matematicheskie Metody, 1976, 12(4):747-756.
[2] NADEZHKINA N, TAKAHASHI W. Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings[J]. Journal of Optimization Theory and Applications, 2006, 128(1):191-201.
[3] TAKAHASHI W, TOYODA M. Weak convergence theorems for nonexpansive mappings and monotone mappings[J]. Journal of Optimization Theory and Applications, 2003, 118(2):417-428.
[4] CENSOR Y, GIBALI A, REICH S. Extensions of Korpelevichs extragradient method for the variational inequality problem in Euclidean space[J]. Optimization, 2010, 61(9):1119-1132.
[5] HE Songnian, YANG Caiping. Solving the variational inequality problem defined on intersection of finite level sets[J]. Abstract and Applied Analysis, 2013, 12(2):94-121.
[6] HE Songnian, XU Hongkun. The supporting hyperplane and an alterntive to solutions of variational inequalities[J]. Journal of Nonlinear and Convex Analysis, 2015, 16(11):2323-2331.
[7] CAI Gang, YEKINI Shehe, IYIOLA Olaniyi Samuel. Iterative algorithms for solving variational inequalities and fixed point problems for asymptotically nonexpansive mappings in Banach spaces[J]. Numerical Algorithms, 2016, 73(3):869-906.
[8] CEGIELSKI A, ZALAS R. Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators[J]. Numerical Functional Analysis and Optimization, 2013, 34(3):255-283.
[9] CRUZ J Y B, IUSEM A N. A strong convergent direct method for monotone variational inequalities in Hilbert spaces[J]. Numerical Functional Analysis and Optimization, 2009, 30(1):23-36.
[10] IUSEM A N, NASRI M. Kropelevichs method for variational inequality problems in Banach spaces[J]. Journal of Global Optimization, 2011, 50(1):59-76.
[11] TANG Guoji, HUANG Nanjing. Kropelevichs method for variational inequality problems on Hadamard manifolds[J]. Journal of Global Optimization, 2012, 54(3):493-509.
[12] TANG Guoji, WANG Xing, LIU Huanwen. A projection-type method for variational inequalities on Hadamard manifolds and verification of solution existence[J]. Optimization, 2013, 64(5):1081-1096.
[13] IUSEM A N, SVAITER B F. A variant of Korpelevichs method for variational inequalities with a new search strategy[J]. Optimization, 1997, 42(2):309-321.
[14] SOLODOV M V, SVAITER B F. A new projection method for variational inequality problems[J]. Siam Journal on Control and Optimization, 1999, 37(3):765-776.
[15] HE Yiran. A new double projection algorithm for variational inequality[J]. Journal of Computational and Applied Mathematics, 2006, 185(1):166-173.
[16] CENSOR Y, GIBALI A, REICH S. The subgradient extragradient method for solving variational inequalities in Hilbert space[J]. Journal of Optimization Theory and Applications, 2011, 148(2):318-335.
[17] 周海云.不动点与零点的迭代方法及其应用[M].北京:国防工业出版社,2016. ZHOU Haiyun. Iterative methods of fixed points and zeros with applications[M]. Beijing: National Defend Industry Press, 2016.
[18] ZHOU Haiyun, ZHOU Yu, FENG Guanghui. Iterative methods for solving a class of monotone variational inequality problems with applications[J]. Journal of Inequalities and Applications, 2015, 2015(68):1-17.
[19] HE Songnian, WU Tao. A modified subgradient extragradient method for solving monotone variational inequalities[J]. Journal of Inequalities and Applications, 2017, 2017(89):1-14.
[20] TAKAHASHI S, TAKAHASHI W. Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces[J]. Journal of Mathematical Analysis and Applications, 2007, 331(1):506-515.
[21] YAO Yonghong, LIOU Yeong Cheng, KANG Shin Min. Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method[J]. Computers and Mathematics with Applications, 2010, 59(11):3472-3480.
[22] MOUDAFI A. Split Monotone Variational inclusions[J]. Journal of Optimization Theory and Applications, 2011, 150(2):275-283.
[23] ANSARI Q H, REHAN A. An iterative method for split hierarchical monotone variational inclusions[J]. Fixed Point Theory and Applications, 2015, 2015(121):1-10.
[24] OPIAL Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings[J]. Bulletin of the American Mathematical Society, 1967, 73(4):591-597.
[25] BROWDER F E. Nonexpansive nonlinear operators in a Banach space[J]. Proceedings of the National Academy of Sciences of the United States of America, 1965, 54(4):1041-1044.
[26] HE Songnian, XU Hongkun. Uniqueness of supporting hyperplanes and an alternative to solutions of variational inequalities[J]. Journal of Global Optimization, 2013, 57(4):1375-1384.
[1]

李爱芹,范丽亚

. 广义内凸性与不变单调性之间的关系[J]. J4, 2008, 43(5): 71-74 .
[2] . 求解变分不等式的修正三步迭代法[J]. J4, 2009, 44(6): 69-74.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
No Suggested Reading articles found!