张量空间上的线性互补问题  

Linear complementarity problem over tensor spaces

作　　者：李夏 黄正海 Xia Li;Zhenghai Huang

机构地区：[1]天津大学数学学院,天津300350

出　　处：《中国科学：数学》2020年第9期1169-1182,共14页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11871051)资助项目。

摘　　要：借助于一类张量收缩积,本文定义一类张量空间上的线性互补问题,简称张量线性互补问题.当所涉及的张量变量退化为向量时,所考虑的问题退化为经典的线性互补问题.对此,首先讨论张量收缩积的一些性质,然后建立张量线性互补问题的理论与算法.具体地,讨论张量线性互补问题的等价模型、可行性与可解性理论、解集的凸性等,提出一个求解张量线性互补问题的外梯度算法,在一定条件下证明算法的收敛性,并给出初步的数值实验结果.With the aid of a contraction product of tensors, we define a class of linear complementarity problems over tensor spaces, which is abbreviated as the tensor linear complementarity problem. When the tensor variable involved is reduced to a vector, the considered problem reduces to the classical linear complementarity problem. In this paper, we discuss certain properties of the contraction product of tensors, and study the theory and algorithm of the concerned problem. Specifically, we first discuss an equivalent model of the problem, the feasibility and solvability of the problem, and convexity of the solution set. Then we propose an extragradient method for solving the tensor linear complementarity problem, and show the convergence of the method under suitable assumptions.The preliminary numerical results are also reported.

关 键 词：张量 张量收缩积 线性互补问题 外梯度法 

分 类 号：O183.2[理学—数学]