Continuity, differentiability and semismoothness of generalized tensor functions

Xia Li; Yong Wang; Zheng-Hai Huang

doi:10.3934/jimo.2020131

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2021, Volume 17, Issue 6: 3525-3550. Doi: 10.3934/jimo.2020131

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Continuity, differentiability and semismoothness of generalized tensor functions

School of Mathematics, Tianjin University, Tianjin 300350, China

^* Corresponding author: Yong Wang
^* Corresponding author: Yong Wang

Received: February 2020

Revised: May 2020

Early access: August 2020

Published: November 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 15A69, 49J50, 49J52.

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