留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

求解非单调变分不等式的一种二次投影算法

王霄婷 龙宪军 彭再云

王霄婷,龙宪军,彭再云. 求解非单调变分不等式的一种二次投影算法 [J]. 应用数学和力学,2022,43(X):1-8 doi: 10.21656/1000-0887.420414
引用本文: 王霄婷,龙宪军,彭再云. 求解非单调变分不等式的一种二次投影算法 [J]. 应用数学和力学,2022,43(X):1-8 doi: 10.21656/1000-0887.420414
Wang Xiaoting, Long Xianjun, Peng Zaiyun. A Double Projection Algorithm for Solving Non-monotone Variational Inequalities.[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420414
Citation: Wang Xiaoting, Long Xianjun, Peng Zaiyun. A Double Projection Algorithm for Solving Non-monotone Variational Inequalities.[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420414

求解非单调变分不等式的一种二次投影算法

doi: 10.21656/1000-0887.420414
基金项目: 国家自然科学基金(11471059);重庆市自然科学基金(cstc2021jcyj-msxmX0721);重庆市教育委员会科学技术研究重点项目(KJZD-K201900801)
详细信息
    作者简介:

    王霄婷(1999—),女,硕士生(E-mail:xiaotingwn@163.com)

    彭再云(1980—),男,教授,博士,博士生导师(通讯作者. E-mail:pengzaiyun@126.com)

  • 中图分类号: O224

A Double Projection Algorithm for Solving Non-monotone Variational Inequalities.

  • 摘要: 投影算法是求解变分不等式问题的主要方法之一。目前,有关投影算法的研究通常需要假设映射是单调且Lipschitz连续的,然而在实际问题中,这些假设条件往往是不满足的。本文利用线搜索方法,提出了一种新的求解非单调变分不等式问题的二次投影算法。在一致连续假设下,证明了算法产生的迭代序列强收敛到变分不等式问题的解。数值实验结果表明了该文所提算法的有效性和优越性。
  • 表  1  $ \varepsilon _{\rm{err}} = 10^{-4}$时不同算法关于维数的比较

    Table  1.   $ \varepsilon _{\rm{err}} = 10^{-4}$ comparison of different algorithms about dimension

    $ x_{1} = (1,1,\cdots ,1)$$ m = 10$$ m = 50$$ m = 100$
    Iter NiCPU time t/sIter NiCPU time t/sIter NiCPU time t/s
    alg 1700.0136890.16861101.0945
    alg 3.3 in ref. [14]478024.0266$ 10^{5}$128.9551$ 10^{5}$755.4308
    alg 4 in ref. [15]$ 10^{5}$12.0259753814.3675148417.3564
    下载: 导出CSV

    表  2  $\varepsilon_ {\rm{err}} = 10^{-4}$时不同算法关于初始点的比较

    Table  2.   $ \varepsilon_{\rm{err}} = 10^{-4}$ Comparison of different algorithms about the initial point

    $ m = 100$$ x_{1} = {\rm{rand}}(100,1)$$ x_{1} = 2*{\rm{rand}}(100,1)$$ x_{1} = 5*{\rm{rand}}(100,1)$
    Iter NiCPU time t/sIter NiCPU time t/sIter NiCPU time t/s
    alg 1930.60471190.73301951.0379
    alg 3.3 in ref. [14]$ 10^{5}$758.4269$ 10^{5}$757.2559$ 10^{5}$748.3109
    alg 4 in ref. [15]4895.46836997.63069249.3821
    下载: 导出CSV

    表  3  $ m = 100$时不同算法关于允许误差的比较

    Table  3.   $ m = 100$ Comparison of different algorithms about allowable error

    $ x_{1} = (1,1,\cdots ,1)$$ 10^{-3}$$ 10^{-5}$$ 10^{-8}$
    Iter NiCPU time t/sIter NiCPU time t/sIter NiCPU time t/s
    alg 1590.01421450.02151661.1689
    alg 3.3 in ref. [14]54460407.0808$ 10^{5}$764.3791$ 10^{5}$751.3783
    alg 4 in ref. [15]6227.1918113313.1318141816.5836
    下载: 导出CSV
  • [1] KINDERLEHRER D, STAMPACCHIA G. An Introduction to Variational Inequalities and Their Applications[M]. New York: Academic Press, 1980.
    [2] FACHINEL F, PANG J S. Finite-Dimensional Variational Inequalities and Complementarity Problems[M]. New York: Springer, 2003.
    [3] HE Y R. A new double projection algorithm for variational inequalities[J]. Journal of Computational and Applied Mathematics, 2006, 185(1): 166-173. doi: 10.1016/j.cam.2005.01.031
    [4] XU H K. Iterative algorithms for nonlinear operators[J]. Journal of the London Mathematical Society, 2002, 66(1): 240-256. doi: 10.1112/S0024610702003332
    [5] HE X, HUANG N J, LI X S. Modified projection methods for solving multi-valued variational inequality without monotonicity[J]. Networks and Spatial Economics, 2019. doi: 10.1007/s11067-019-09485-2
    [6] 贺月红, 龙宪军. 求解伪单调变分不等式问题的惯性收缩投影算法[J]. 数学物理学报, 2021, 41A(6): 1897-1911. (HE Yuehong, LONG Xianjun. A inertial contraction and projection algorithm for pseudomonotone variational inequalities problems[J]. Acta Mathematica Scientia, 2021, 41A(6): 1897-1911.(in Chinese) doi: 10.3969/j.issn.1003-3998.2021.06.026

    HE Yuehong, LONG Xianjun. A inertial contraction and projection algorithm for pseudomonotone variational inequalities problems[J]. Acta Mathematica Scientia, 2021, 41A(6): 1897-1911. (in Chinese) doi: 10.3969/j.issn.1003-3998.2021.06.026
    [7] 万升联. 解变分不等式的一种二次投影算法[J]. 数学物理学报, 2021, 41A(1): 237-244. (WAN Shenglian. A double projection algorithm for solving variational inequalities[J]. Acta Mathematica Scientia, 2021, 41A(1): 237-244.(in Chinese) doi: 10.3969/j.issn.1003-3998.2021.01.019

    WAN Shenglian. A double projection algorithm for solving variational inequalities[J]. Acta Mathematica Scientia, 2021, 41A(1): 237-244. (in Chinese) doi: 10.3969/j.issn.1003-3998.2021.01.019
    [8] 杨军. 非单调变分不等式黄金分割算法研究[J]. 应用数学和力学, 2021, 42(7): 764-770. (YANG Jun. A golden ratio algorithm for solving nonmonotone variational inequalities[J]. Applied Mathematics and Mechanics, 2021, 42(7): 764-770.(in Chinese)

    YANG Jun. A golden ratio algorithm for solving nonmonotone variational inequalities[J]. Applied Mathematics and Mechanics, 2021, 42(7): 764-770. (in Chinese)
    [9] YE M L, HE Y R. A double projection method for solving variational inequalities without monotonicity[J]. Computational Optimization and Applications, 2015, 60(1): 141-150. doi: 10.1007/s10589-014-9659-7
    [10] HE B S. A class of projection and contraction methods for monotone variational inequalities[J]. Applied Mathematics and Optimization, 1997, 35: 69-76. doi: 10.1007/s002459900037
    [11] GOLDSTEIN A. Convex programming in Hilbert space[J]. Bulltin of the American Mathematical Society, 1964, 70(5): 709-710. doi: 10.1090/S0002-9904-1964-11178-2
    [12] KORPELEVICH G M. The extragradient method for finding saddle points and other problems[J]. Ekonomika I Matematicheskie Metody, 1976, 12: 747-756.
    [13] 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. doi: 10.1137/S0363012997317475
    [14] VUONG P T, SHEHU Y. Convergence of an extragradient-type method for variational inequality with applications to optimal control problems[J]. Numerical Algorithms, 2019, 81(1): 269-291. doi: 10.1007/s11075-018-0547-6
    [15] REICH S, THONG D V, DONG Q L, et al. New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings[J]. Numerical Algorithms, 2021, 87(2): 527-549. doi: 10.1007/s11075-020-00977-8
    [16] HE Y R. Solvability of the minty variational inequality[J]. Journal of Optimization Theory and Applications, 2017, 174(3): 686-692. doi: 10.1007/s10957-017-1124-1
  • 加载中
表(3)
计量
  • 文章访问数:  14
  • HTML全文浏览量:  10
  • PDF下载量:  3
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-12-31
  • 修回日期:  2022-03-23
  • 网络出版日期:  2022-06-21

目录

    /

    返回文章
    返回