求解单调包含问题的惯性混合非精确邻近点算法

删除或更新信息，请邮件至freekaoyan#163.com(#换成@)

本站小编 Free考研考试/2021-12-27

求解单调包含问题的惯性混合非精确邻近点算法何明明, 彭建文重庆师范大学数学科学学院, 重庆 401331 Inertial Hybrid Inexact Proximal Point Method for Solving Monotone Inclusions Problems HE Mingming, PENG JianwenSchool of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

摘要
图/表
参考文献
相关文章(15)
点击分布统计
下载分布统计
-->

全文: PDF(600 KB) HTML (1 KB)
输出: BibTeX | EndNote (RIS)

摘要本文提出了求解单调包含问题的一类新的惯性混合非精确邻近点算法（简记为iHIPPA）.在适当的参数假设下，我们证明了求解单调包含问题的iHIPPA所产生点列的弱收敛性，获得了iHIPPA的非渐近收敛率为O（1/√k）及iHIPPA的遍历迭代复杂性为O（1/k）.作为应用，我们还建立了求解单调变分包含问题的惯性邻近收缩算法，求解广义变分不等式问题的惯性投影邻近点算法，及求解原始—对偶问题的惯性非精确调比部分逆算法产生点列的收敛性及相应算法的非渐近收敛率及遍历迭代复杂性.本文结果推广和改进了文献中的相应结论.最后，本文应用新的惯性交替方向乘子法用以求解LASSO问题，而且一些初步的试验结果表明了新的算法的优越性.

服务

加入引用管理器

E-mail Alert

RSS

收稿日期: 2019-06-25

PACS:

O221.2

基金资助:国家自然科学基金重大项目（11991024），国家自然科学基金面上项目（11171363）和重庆市基础科学与前沿技术研究专项重点项目（cstc2015jcyjBX0029）资助.

引用本文:

何明明, 彭建文. 求解单调包含问题的惯性混合非精确邻近点算法[J]. 应用数学学报, 2020, 43(4): 700-727. HE Mingming, PENG Jianwen. Inertial Hybrid Inexact Proximal Point Method for Solving Monotone Inclusions Problems. Acta Mathematicae Applicatae Sinica, 2020, 43(4): 700-727.

链接本文:

http://123.57.41.99/jweb_yysxxb/CN/或 http://123.57.41.99/jweb_yysxxb/CN/Y2020/V43/I4/700

[1]	Boţ R I, Csetnek E R. A hybrid proximal-extragradient algorithm with inertial effects. Numerical Functional Analysis and Optimization, 2015, 36(8):951-963
[2]	Martinet B. Régularisation d'inéquations variationnelles par approximations successives. ESAIM:Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 1970, 4(R3):154-158
[3]	Rockafellar R T. Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization, 1976, 14(5):877-898
[4]	Solodov M V, Svaiter B F. A hybrid projection-proximal point algorithm. Journal of Convex Analysis, 1999, 6(1):59-70
[5]	Solodov M V, Svaiter B F. A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator. Set-Valued Analysis, 1999, 7:323-345
[6]	Burachik R S, Iusem A, Svaiter B F. Enlargement of monotone operators with applications to variational inequalities. Set-Valued Analysis, 1997, 5(2):159-180
[7]	Tseng P. A modified forward-backward splitting method for maximal monotone mappings. SIAM Journal on Control and Optimization, 2000, 38:431-446
[8]	Solodov M V, Svaiter B F. A unified framework for some inexact proximal point algorithms. Numerical Functional Analysis and Optimization, 2001, 22:1013-1035
[9]	Solodov M V. A class of decomposition methods for convex optimization and monotone variational inclusions via the hybrid inexact proximal point framework. Optimization Methods and Software, 2004, 19:557-575
[10]	Chen G, Teboulle M. A proximal-based decomposition method for convex minimization problems. Mathematical Programming, 1994, 64:81-101
[11]	Parente L A, Lotito P A, Solodov M V. A class of inexact variable metric proximal point algorithms. SIAM Journal on Optimization, 2008, 19:240-260
[12]	Lotito P A, Parente L A, Solodov M V. A class of variable metric decomposition methods for monotone variational inclusions. Journal of Convex Analysis, 2009, 16:857-880
[13]	Chen C, Chan R H, Ma S, Yang J. Inertial proximal ADMM for linearly constrained separable convex optimization. SIAM Journal on Imaging Sciences, 2015, 8(4):2239-2267
[14]	Boţ R I, Csetnek E R, Hendrich C. Inertial Douglas-Rachford splitting for monotone inclusion problems. Applied Mathematics and Computation, 2015, 256:472-487
[15]	Moudafi A. A hybrid inertial projection-proximal method for variational inequalities. Journal of Inequalities in Pure & Applied Mathematics, 2004, 5(3):1-13
[16]	Burachik R S, Svaiter B F. ε-enlargements of maximal monotone operators in Banach spaces. SetValued Analysis, 1999, 7(2):117-132
[17]	Burachik R S, Sagastizábal C A, Svaiter B F. ε-enlargements of maximal monotone operators:theory and applications. In:M. Fukushima, L. Qi (eds), Reformulation-Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Acad. Publ., 1999, 25-44
[18]	Bauschke H H, Combettes P L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. New York:Springer-Verlag, 2010
[19]	Opial Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society, 1967, 73(4):591-597
[20]	Alvarez F, Attouch H. An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Analysis, 2001, 9:3-11
[21]	Chen C, Ma S, Yang J. A general inertial proximal point algorithm for mixed variational inequality problem. SIAM Journal on Optimization, 2015, 25(4):2120-2142
[22]	Moudafi A, Oliny M. Convergence of a splitting inertial proximal method for monotone operators. Journal of Computational and Applied Mathematics, 2003, 155:447-454
[23]	Attouch H, Cabot A. Convergence of a relaxed inertial proximal algorithm for maximally monotone operators. Preprint hal-01708905, 2018.
[24]	Alvarez F. Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM Journal on Optimization, 2003, 14(3):773-782
[25]	Alves M M, Marcavillaca R T. On inexact relative-error hybrid proximal extragradient, forwardbackward and Tseng's modified forward-backward methods with inertial effects. Set-Valued and Variational Analysis, 2019:1-25 https://doi.org/10.1007/s11228-019-00510-7.
[26]	王艺楠. Hilbert空间中单调变分包含问题的算法研究. 天津:中国民航大学, 2018(Wang Y. The iterative algorithm of monotone variational inclusion problem in Hilbert space. Tianjin:Civil Aviation University of China, 2018)
[27]	Zhang C, Wang Y. Proximal algorithm for solving monotone variational inclusion. Optimization, 2018(1):1-13
[28]	Xia F Q, Huang N J. A projection-proximal point algorithm for solving generalized variational inequalities. Journal of Optimization Theory and Applications, 2011, 150:98-117
[29]	Briceño A, Luis M. Forward-Douglas-Rachford splitting and forward-partial inverse method for solving monotone inclusions. Optimization, 2015, 64(5):1239-1261
[30]	Spingarn J E. Partial inverse of a monotone operator. Applied Mathematics and Optimization, 1983, 10(3):247-265
[31]	Burachik R S, Sagastizábal C A, Scheimberg S. An inexact method of partial inverses and a parallel bundle method. Optimization Methods and Software, 2006, 21(3):385-400
[32]	Adona V A, Gonçalves M L N, Melo J G. A partially inexact proximal alternating direction method of multipliers and its iteration-complexity analysis. Journal of Optimization Theory and Applications, 2019, 182(2):640-666
[33]	Tibshirani R. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society:Series B (Methodological), 1996, 58(1):267-288
[34]	Boyd S, Parikh N, Chu E, et al. Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends. Mach. Learn., 2011, 3:1-122

[1]	黎健玲, 王培培, 曾友芳, 简金宝. 凸二次半定规划一个长步原始对偶路径跟踪算法[J]. 应用数学学报, 2020, 43(1): 12-32.
[2]	李成进, 张圣贵, 吴慧慧. 广义半正定最小二乘问题的近似点迭代法[J]. 应用数学学报, 2019, 42(3): 371-384.
[3]	王福胜, 高娟, 赵媛璐, 姜合峰. 混合约束Minimax问题的基于序列线性方程组的模松弛SQP算法[J]. 应用数学学报, 2019, 42(2): 242-253.
[4]	李明强, 韩丛英, 郭田德. 新的梯度算法求解单位球笛卡尔积约束优化问题[J]. 应用数学学报, 2018, 41(1): 43-54.
[5]	张艳君, 赵金玲, 徐尔. 求解多集分裂可行问题的一种新的松弛投影算法[J]. 应用数学学报, 2017, 40(5): 641-652.
[6]	唐国吉, 汪星, 叶明露. 混合变分不等式的一个投影型方法[J]. 应用数学学报, 2016, 39(4): 574-585.
[7]	刘美杏, 唐春明, 简金宝. 不等式约束优化基于新型积极识别集的SQCQP算法[J]. 应用数学学报, 2015, 38(2): 222-234.
[8]	陈智勤, 李成进, 张圣贵. 物流管理中的图分割问题[J]. 应用数学学报, 2015, 38(2): 356-365.
[9]	马国栋, 简金宝, 江羡珍. 一个具有下降性的改进Fletcher-Reeves共轭梯度法[J]. 应用数学学报, 2015, 38(1): 89-97.
[10]	张长温, 李师正. 凸半无限规划正则性的刻画[J]. 应用数学学报(英文版), 2014, 37(4): 735-744.
[11]	黎健玲, 黄仁帅, 简金宝. 非线性互补约束优化一个全局收敛的QP-free算法[J]. 应用数学学报(英文版), 2014, 37(4): 629-644.
[12]	裴永刚, 靳利, 申培萍. 求解带指数凹多乘积规划问题的对偶界方法[J]. 应用数学学报(英文版), 2013, (1): 115-125.
[13]	王永丽, 韩丛英, 李田, 李明强. 求解不等式约束优化问题无严格互补松弛条件的one QP-Free新算法[J]. 应用数学学报(英文版), 2013, (1): 1-13.
[14]	吴秋峰, 刘振忠. Stiefel流形上的梯度下降法[J]. 应用数学学报(英文版), 2012, (4): 719-727.
[15]	王秀玉, 姜兴武, 刘庆怀. 非线性互补问题的组合同伦算法 [J]. 应用数学学报(英文版), 2012, (3): 430-440.

PDF全文下载地址:

http://123.57.41.99/jweb_yysxxb/CN/article/downloadArticleFile.do?attachType=PDF&id=14797