非负象限权互补问题的免导数非单调光滑牛顿法  

A derivative-free nonmonotone smoothing Newton method for weighted complementarity problem over nonnegative quadrant

作　　者：刘文丽 迟晓妮[1,2,3] 张璐 李绍刚 LIU Wenli;CHI Xiaoni;ZHANG Lu;LI Shaogang(School of Mathematics and Computing Science,Guilin University of Electronic Technology,Guilin 541004,China;Guangxi Key Laboratory of Autom atic Detection Technology and Instrument,Guilin University of Electronic Technology,Guilin 541004,China;Guangxi Key Laboratory of Cryptography and Information Security,Guilin University of Electronic Technology,Guilin 541004,China)

机构地区：[1]桂林电子科技大学数学与计算科学学院,广西桂林541004 [2]桂林电子科技大学广西自动检测技术与仪器重点实验室,广西桂林541004 [3]桂林电子科技大学广西密码学与信息安全重点实验室,广西桂林541004

出　　处：《桂林电子科技大学学报》2021年第6期504-509,共6页Journal of Guilin University of Electronic Technology

基　　金：国家自然科学基金(11861026);广西自然科学基金(2021GXNSFAA220034);广西密码学与信息安全重点实验室基金(GCIS201819);广西自动检测技术与仪器重点实验室基金(YQ18112)。

摘　　要：光滑牛顿法是求解互补问题最常用的方法,故将非单调光滑牛顿法推广到求解非负象限权互补问题上。首先,构造新的权互补问题的光滑函数,并研究其连续性、可微性等性质;其次,基于该函数,将非负象限权互补问题转化成含光滑参数的光滑方程组,当光滑参数为0时,该方程组的解即为非负象限权互补问题的解;最后,借助光滑方程组的连续性、雅可比矩阵非奇异性等性质,提出一种求解该方程组的非单调光滑牛顿法。为使求解算法高效稳定,所提算法采用新的免导数非单调线搜索技术。在适当假设下,证明了算法全局收敛性质。利用算法求解非负象限线性权互补问题和非负象限非线性权互补问题,验证了算法的有效性和稳定性.The smoothing Newton method is the most popular method to solve the complementarity problem.Therefore,the nonmonotone smoothing Newton method is extended to solve the weighted complementarity problem over nonnegative quadrant.Firstly,we construct a new smoothing function of the weighted complementarity problem,and study its properties,such as continuity and differentiability.Secondly,based on the smoothing function,the weighted complementarity problem over nonnegative quadrant is transformed into a system of equations with the smoothing parameter.When the smoothing parameter is zero,the solution of the equations is the solution of the weighted complementarity problem over nonnegative quadrant.Next,by continuity and nonsingularity of Jacobian matrix of the smoothing equations,a nonmonotone smoothing Newton method for solving the equations is proposed.To guarantee that the method is efficient and stable,a derivative-free nonmonotone line search is used in the method.Under suitable assumption,the global convergence of the method is proved.Finally,by solving both the linear weighted complementarity problems and the nonlinear weighted complementarity problems over nonnegative quadrant,the effectiveness and stability of the method are verified.

关 键 词：权互补问题 光滑函数 免导数非单调线搜索 光滑牛顿法 全局收敛性 

分 类 号：O221[理学—运筹学与控制论]