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解一类结构变分不等式问题的非精确并行交替方向法

冯俊锴1,*  张海斌1 秦嫒1  张凯丽1   

  1. 1. 北京工业大学应用数理学院, 北京 100124
  • 收稿日期:2018-02-20 出版日期:2018-06-15 发布日期:2018-06-15
  • 通讯作者: 冯俊锴 fengjunkai1989@126.com
  • 基金资助:

    国家自然科学基金(No.11771003)

An inexact parallel alternating direction method for structured variational inequalities

FENG Junkai1,*  ZHANG Haibin1  QIN Yuan1  ZHANG Kaili1   

  1. 1. College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
  • Received:2018-02-20 Online:2018-06-15 Published:2018-06-15

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摘要：

带线性约束的具有两分块结构的单调变分不等式问题, 出现在许多现代应用中, 如交通和经济问题等. 基于该问题良好的可分结构, 分裂型算法被广泛研究用于其求解. 提出新的带回代的非精确并行交替方向法解该类问题, 在每一步迭代中，首先以并行模式通过投影得到预测点, 然后对其校正得到下一步的迭代点. 在压缩型算法的理论框架下, 在适当条件下证明了所提算法的全局收敛性. 数值结果表明了算法的有效性. 此外, 该算法可推广到求解具有多分块结构的问题.

关键词: 变分不等式, 交替方向法, 并行算法, 预测-校正法

Abstract:

This paper considers the monotone variational inequality problems with two separable blocks subject to linear coupling constraints. Problems of this type arise in many contemporary applications including traffic assignment and economics. Based on its favorable separable structure, splitting type methods have been studied. In this paper, we introduce a new inexact parallel alternating direction method with a substitution to solve this family of problems. At each iteration, one can get a predictor by using projection in  parallel fashion, then corrects the predictor to generate the new iterate. For the proposed algorithm, we prove its convergence under mild conditions via the analytic framework of contractive type methods. Some numerical results  are reported to support the efficiency of the new method. Moreover,  the proposed method can be extended to solve the variational inequality problems with multi-blocks.

Key words: variational inequalities, alternating direction methods, parallel methods, prediction-correction methods