Open AccessContinuity, differentiability and semismoothness of generalized tensor functions
Author(s) -
Xia Li,
Yong Wang,
Zheng-Hai Huang
Publication year - 2021
Publication title -
journal of industrial and management optimization
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.325
H-Index - 32
eISSN - 1553-166X
pISSN - 1547-5816
DOI - 10.3934/jimo.2020131
Subject(s) - tensor contraction , tensor product of hilbert spaces , mathematics , differentiable function , tensor (intrinsic definition) , pure mathematics , tensor product , symmetric tensor , lipschitz continuity , cartesian tensor , scalar (mathematics) , tensor field , tensor density , mathematical analysis , algebra over a field , exact solutions in general relativity , geometry
A large number of real-world problems can be transformed into mathematical problems by means of third-order real tensors. Recently, as an extension of the generalized matrix function, the generalized tensor function over the third-order real tensor space was introduced with the aid of a scalar function based on the T-product for third-order tensors and the tensor singular value decomposition; and some useful algebraic properties of the function were investigated. In this paper, we show that the generalized tensor function can inherit a lot of good properties from the associated scalar function, including continuity, directional differentiability, Fréchet differentiability, Lipschitz continuity and semismoothness. These properties provide an important theoretical basis for the studies of various mathematical problems with generalized tensor functions, and particularly, for the studies of tensor optimization problems with generalized tensor functions.