Further study on tensor absolute value equations
- PDF / 302,831 Bytes
- 20 Pages / 612 x 792 pts (letter) Page_size
- 63 Downloads / 230 Views
. ARTICLES .
https://doi.org/10.1007/s11425-018-9560-3
Further study on tensor absolute value equations Chen Ling1 , Weijie Yan1 , Hongjin He1,∗ & Liqun Qi2 1Department
of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China; of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China
2Department
Email: [email protected], [email protected], [email protected], [email protected] Received November 3, 2018; accepted June 28, 2019
Abstract
In this paper, we consider the tensor absolute value equations (TAVEs), which is a newly introduced
problem in the context of multilinear systems. Although the system of the TAVEs is an interesting generalization of matrix absolute value equations (AVEs), the well-developed theory and algorithms for the AVEs are not directly applicable to the TAVEs due to the nonlinearity (or multilinearity) of the problem under consideration. Therefore, we first study the solutions existence of some classes of the TAVEs with the help of degree theory, in addition to showing, by fixed point theory, that the system of the TAVEs has at least one solution under some checkable conditions. Then, we give a bound of solutions of the TAVEs for some special cases. To find a solution to the TAVEs, we employ the generalized Newton method and report some preliminary results. Keywords
tensor absolute value equations, H+ -tensor, P-tensor, copositive tensor, generalized Newton
method MSC(2010)
15A48, 15A69, 65K05, 90C30, 90C20
Citation: Ling C, Yan W J, He H J, et al. Further study on tensor absolute value equations. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-018-9560-3
1
Introduction
The system of the absolute value equations (AVEs) investigated in literature is given by Ax − |x| = b,
(1.1)
where A ∈ Rn×n , b ∈ Rn , and |x| denotes the vector with absolute values of each component of x. The importance of the AVEs (1.1) has been well documented in the monograph [6] due to its equivalence to the classical linear complementarity problems. More generally, Rohn [34] introduced the following problem: Ax + B|x| = b,
(1.2)
where A, B ∈ Rm×n and b ∈ Rm . Apparently, (1.2) covers (1.1) with the setting of B being a negative identity matrix. In what follows, we also call such a general problem (1.2) a system of the AVEs for simplicity. Since the seminal work [24] investigated the existence and nonexistence of solutions to the system of the AVEs (1.1) in 2006, the system of the AVEs has been studied extensively by many researchers. In the past decade, a series of interesting theoretical results including NP-hardness [22, 24], solvability * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019 ⃝
math.scichina.com
link.springer.com
Ling C et al.
2
Sci China Math
[24, 34], and equivalent reformulations [22, 24, 27] of the system of the AVEs have been developed. Also, many efficient algorithms have been designed to solve the system of the AVEs (see, e.g., [3,13,14,23,25,44] and the re
Data Loading...
