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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 ›› 2018, Vol. 35 ›› Issue (6): 673-683.doi: 10.3969/j.issn.1005-3085.2018.06.007

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求解广义Lyapunov方程的非单调谱投影梯度法

喻思婷,   李春梅,   段雪峰   

  1. 桂林电子科技大学数学与计算科学学院， 桂林  541004
  • 收稿日期:2016-12-01 接受日期:2017-04-25 出版日期:2018-12-15 发布日期:2019-02-15
  • 基金资助:
    国家自然科学基金(11561015; 11761024)；广西自然科学基金(2016GXNSFFA380009; 2016GXNSFAA380074; 2017GXNSFBA198082)；广西密码学与信息安全重点实验室开放基金(GCIS201616).

Nonmonotone Spectral Projection Gradient Method for Solving Generalized Lyapunov Equations

YU Si-ting,   LI Chun-mei,   DUAN Xue-feng   

  1. School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004
  • Received:2016-12-01 Accepted:2017-04-25 Online:2018-12-15 Published:2019-02-15
  • Supported by:
    The National Natural Science Foundation of China (11561015; 11761024); the Natural Science Foundation of Guangxi Province (2016GXNSFFA380009; 2016GXNSFAA380074; 2017GXNSFBA198082); the Guangxi Key Laboratory of Cryptography and Information Security (GCIS201616).

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摘要： 本文研究双线性控制系统中的一类广义Lyapunov方程的半正定解．基于凸函数的局部极小解就是全局极小解这一良好性质，首先将广义Lyapunov方程的半正定解问题等价转化为凸优化问题．利用非单调线搜索技术确定步长，构造了非单调谱投影梯度方法求解这一等价问题．最后用数值例子验证了新方法的可行性和有效性．

关键词: 广义Lyapunov方程, 半正定解, 非单调线搜索, 投影梯度方法

Abstract: In this paper, we consider the positive semidefinite solution to a class of generalized Lyapunov matrix equation, which arises in bilinear systems. Based on the good property that the local minimizer of a convex function is also the global minimizer, the positive semidefinite solution of the generalized Lyapunov equation is transformed into a convex optimization problem. By using the nonmonotone line search technique, we develop a nonmonotone spectral projected gradient method to solve this equivalent problem. Finally, numerical examples are presented to illustrate the feasibility and effectiveness of the new method.

Key words: generalized Lyapunov equation, positive semidefinite solution, nonmonotone line search, projected gradient method

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