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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2017, Vol. 34 ›› Issue (5): 507-516.doi: 10.3969/j.issn.1005-3085.2017.05.006

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求解变分不等式的非单调混合Newton算法

宫恩龙1,   王宣战2,   高苗苗2,   杜小雨2,   孙清滢2   

  1. 1- 青岛酒店管理职业技术学院，青岛  266100
    2- 中国石油大学（华东）理学院，青岛  266580
  • 收稿日期:2015-12-24 接受日期:2016-08-05 出版日期:2017-10-15 发布日期:2017-12-15
  • 基金资助:
    国家自然科学基金(61201455).

A Non-monotone Hybrid Newton Method for Solving the Variational Inequality Problems

GONG En-long1,   WANG Xuan-zhan2,   GAO Miao-miao2,   DU Xiao-yu2,   SUN Qing-ying2   

  1. 1- Qingdao Hotel Management College, Qingdao 266100
    2- College of Science, China University of Petroleum (Huadong), Qingdao 266580
  • Received:2015-12-24 Accepted:2016-08-05 Online:2017-10-15 Published:2017-12-15
  • Supported by:
    The National Natural Science Foundation of China (61201455).

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摘要： 本文运用广义$D$-间隙函数可以将变分不等式问题转化为一个无约束最优化问题，即极小化广义$D$-间隙函数的一般形式$g_{\alpha\beta} $，基于非单调线搜索技术提出一种非单调混合Newton算法，并给出了算法的全局收敛性分析．在适当条件下，证明了算法具有全局二次收敛性．同时在映射$F$强单调但不需要Lipschitz连续的情况下，为算法提供了一个全局误差界．数值结果表明新算法是有效的．

关键词: 广义$D$-间隙函数, 非单调线搜索, 全局收敛, 全局误差界

Abstract: In this paper, the variational inequality problem is transformed as an unconstrained optimization problem through the generalized $D$-gap function. A non-monotone hybrid Newton method based on Zhang H.C.'s non-monotone line search technique is proposed for minimizing the general form of the generalized $D$-gap function. Then, the global convergence property of the algorithm is analyzed. Under some proper conditions, we prove that the algorithm is globally quadratically convergent. Moreover, we obtain a global error bound of the algorithm when the mapping $F$ is strongly monotone without Lipschitz continuous. Numerical results indicate that the new algorithm is efficient.

Key words: generalized $D$-gap function, non-monotone line search, global convergence, global error bound

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