求解拟单调变分不等式问题与不动点问题公共解的新投影算法

摘要/Abstract

图/表 8

参考文献 31

Metrics

摘要：

该文在 Hilbert 空间中提出具有惯性项的 Tseng 型外梯度算法, 找到了拟单调变分不等式问题与半压缩映射的不动点问题的公共解. 在拟单调和一致连续的条件下, 获得了算法所生成序列的强收敛性. 最后, 通过一些数值例子说明了该算法的有效性.

关键词: 变分不等式问题与不动点问题, Tseng 型外梯度算法, 拟单调映射, 半压缩映射, 强收敛

Abstract:

In this paper, we introduce a new inertial Tseng's extragradient algorithm for finding a common element of the set of solutions of a quasimonotone variational inequality problem and the set of fixed points of a demicontractive mapping in real Hilbert spaces. The strong convergence of the algorithm is proved under quasimonotone and uniformly continuous assumptions of the variational inequality mapping. Finally, some numerical examples illustrate the effectiveness of our algorithm.

Key words: variational inequality problem and fixed point problem, Tseng's extragradient algorithm, quasimonotone mapping, demicontractive mapping, strong convergence

中图分类号:

O224

王吴静, 朱美玲, 张永乐. 求解拟单调变分不等式问题与不动点问题公共解的新投影算法[J]. 数学物理学报, 2025, 45(1): 236-255.

Wujing Wang, Meiling Zhu, Yongle Zhang. A New Projection Algorithm for Solving Quasimonotone Variational Inequality Problems and Fixed Point Problems[J]. Acta mathematica scientia,Series A, 2025, 45(1): 236-255.

表 1

图1

例 4.1 中 $k=10000, \varepsilon=10^{-12}$ 的数值实验结果"

图1

表 2

例 4.2 的初始点为 $x^0(t)=t+0.5$ cost 的数值实验结果"

表 2

图2

例 4.2 的初始点为 $x^0(t)=t+0.5 cost, \varepsilon=10^{-13}$ 的数值实验结果"

图2

表 3

图3

例 4.3的初始点为 $x^0=(-5,-5)^T, \varepsilon=10^{-6}$ 的数值实验结果"

图3

表 4

图4

例 4.4 中初始点 $x^0=x^1=100, \varepsilon=10^{-10}$ 的数值实验结果"

图4

[1]	Iiduka H. Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping. Mathematical Programming, 2015, 149(1): 131-165
[2]	Maingé P E. A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM Journal on Control and Optimization, 2008, 47(3): 1499-1515
[3]	Thong D V, Hieu D V. Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems. Optimization, 2018, 67(1): 83-102
[4]	Nadezhkina N, Takahashi W. Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications, 2006, 128: 191-201
[5]	Takahashi W, Toyoda M. Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications, 2003, 118: 417-428
[6]	Thong D V, Hieu D V. New extragradient methods for solving variational inequality problems and fixed point problems. Journal of Fixed Point Theory and Applications, 2018, 20(3): Article 129
[7]	Thong D V, Hieu D V. Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems. Numerical Algorithms, 2019, 82(3): 761-789
[8]	Thong D V, Liu L L, Dong Q L, et al. Fast relaxed inertial Tseng's method-based algorithm for solving variational inequality and fixed point problems in Hilbert spaces. Journal of Computational and Applied Mathematics, 2023, 418: 114739
[9]	段洁, 夏福全. 求解变分不等式与不动点问题公共解的新 Tseng 型外梯度算法. 数学物理学报, 2023, 43A(1): 274-290
	Duan J, Xia F Q. A new Tseng-like extragradient algorithm for common solutions of variational inequalities and fixed point problems. Acta Math Sci, 2023, 43A(1): 274-290
[10]	Goldstein A A. Convex programming in Hilbert space. Bulletin of the American Mathematical Society, 1964, 70(5): 109-112
[11]	Mann W R. Mean value methods in iteration. Proceedings of the American Mathematical Society, 1953, 4(3): 506-510
[12]	Korpelevich G M. The extragradient method for finding saddle points and other problems. Matecon, 1976, 12: 747-756
[13]	Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. Journal of Optimization Theory and Applications, 2011, 148(2): 318-335 pmid: 21490879
[14]	Tseng P. A modified forward-backward splitting method for maximal monotone mappings. SIAM Journal on Control and Optimization, 2000, 38(2): 431-446
[15]	Halpern B. Fixed points of nonexansive maps. Bulletin of the American Mathematical Society, 1967, 73: 957-961
[16]	Wang Z B, Chen X, Yi J, Chen Z Y. Inertial projection and contraction algorithms with larger step sizes for solving quasimonotone variational inequalities. Journal of Global Optimization, 2022, 82(3): 499-522
[17]	Yin T C, Wu Y K, Wen C F. An iterative algorithm for solving fixed point problems and quasimonotone variational inequalities. Journal of Mathematics, 2022, 2022(1): 8644675
[18]	Chidume C E, Măuşter Ş. Iterative methods for the computation of fixed points of demicontractive mappings. Journal of Computational and Applied Mathematics, 2010, 234(3): 861-882
[19]	Mongkolkeha C, Cho Y J, Kumam P. Convergence theorems for k-dimeicontactive mappings in Hilbert spaces. Mathematical Inequalities & Applications, 2013, 16(4): 1065-1082
[20]	Kraikaew R, Saejung S. Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. Journal of Optimization Theory and Applications, 2014, 163: 399-412
[21]	Tian M, Xu G. Inertial modified Tseng's extragradient algorithms for solving monotone variational inequalities and fixed point problems. J Nonlinear Funct Anal, 2020, 2020: Article 35
[22]	Bauschke H H, Combettes P L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. New York: Springer, 2011
[23]	Denisov S V, Semenov V V, Chabak L M. Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybernetics and Systems Analysis, 2015, 51: 757-765
[24]	Saejung S, Yotkaew P. Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications, 2012, 75(2): 742-750
[25]	Ye M L. An infeasible projection type algorithm for nonmonotone variational inequalities. Numerical Algorithms, 2022, 89(4): 1723-1742
[26]	Ye M L, He Y R. A double projection method for solving variational inequalities without monotonicity. Computational Optimization and Applications, 2015, 60(1): 141-150
[27]	Iusem A, Otero R G. Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces. Numerical Functional Analysis and Optimization, 2001, 22(5/6): 609-640
[28]	Cai G, Dong Q L, Peng Y. Strong convergence theorems for solving variational inequality problems with pseudo-monotone and non-Lipschitz operators. Journal of Optimization Theory and Applications, 2021, 188: 447-472
[29]	Shehu Y, Dong Q L, Jiang D. Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization, 2019, 68(1): 385-409
[30]	Rehman H U, Kumam P, Kumam W, Sombut K. A new class of inertial algorithms with monotonic step sizes for solving fixed point and variational inequalities. Mathematical Methods in the Applied Sciences, 2022, 45(16): 9061-9088
[31]	杨静, 龙宪军. 关于伪单调变分不等式与不动点问题的新投影算法. 数学物理学报, 2022, 42A(3): 904-919
	Yang J, Long X J. A new projection algorithm for solving pseudo-monotone variational inequality and fixed point problems. Acta Math Sci, 2022, 42A(3): 904-919

Just accepted

Online first

Just accepted

Online first

Viewed

Full text

	From	local

	Times	53
	Rate	100%

Abstract

Just accepted	Online first	Issue

0	0	36

From	Others	local

Times	33	3
Rate	92%	8%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed