# American Institute of Mathematical Sciences

November  2021, 17(6): 3525-3550. doi: 10.3934/jimo.2020131

## Continuity, differentiability and semismoothness of generalized tensor functions

 School of Mathematics, Tianjin University, Tianjin 300350, China

* Corresponding author: Yong Wang

Received  February 2020 Revised  May 2020 Published  November 2021 Early access  August 2020

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.

Citation: Xia Li, Yong Wang, Zheng-Hai Huang. Continuity, differentiability and semismoothness of generalized tensor functions. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3525-3550. doi: 10.3934/jimo.2020131
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##### References:
 [1] H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549 [2] Jinghong Liu, Yinsuo Jia. Gradient superconvergence post-processing of the tensor-product quadratic pentahedral finite element. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 495-504. doi: 10.3934/dcdsb.2015.20.495 [3] Jin Wang, Jun-E Feng, Hua-Lin Huang. Solvability of the matrix equation $AX^{2} = B$ with semi-tensor product. Electronic Research Archive, 2021, 29 (3) : 2249-2267. doi: 10.3934/era.2020114 [4] Jan Boman, Vladimir Sharafutdinov. Stability estimates in tensor tomography. Inverse Problems & Imaging, 2018, 12 (5) : 1245-1262. doi: 10.3934/ipi.2018052 [5] Shenglong Hu. A note on the solvability of a tensor equation. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021146 [6] Yuning Liu, Wei Wang. On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115 [7] Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, 2021, 15 (3) : 475-498. doi: 10.3934/ipi.2021001 [8] Mengmeng Zheng, Ying Zhang, Zheng-Hai Huang. Global error bounds for the tensor complementarity problem with a P-tensor. Journal of Industrial & Management Optimization, 2019, 15 (2) : 933-946. doi: 10.3934/jimo.2018078 [9] Yiju Wang, Guanglu Zhou, Louis Caccetta. Nonsingular $H$-tensor and its criteria. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1173-1186. doi: 10.3934/jimo.2016.12.1173 [10] Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013 [11] Henry O. Jacobs, Hiroaki Yoshimura. Tensor products of Dirac structures and interconnection in Lagrangian mechanics. Journal of Geometric Mechanics, 2014, 6 (1) : 67-98. doi: 10.3934/jgm.2014.6.67 [12] François Monard. Efficient tensor tomography in fan-beam coordinates. Inverse Problems & Imaging, 2016, 10 (2) : 433-459. doi: 10.3934/ipi.2016007 [13] Liqun Qi, Shenglong Hu, Yanwei Xu. Spectral norm and nuclear norm of a third order tensor. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021010 [14] Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049 [15] Zhong Wan, Chunhua Yang. New approach to global minimization of normal multivariate polynomial based on tensor. Journal of Industrial & Management Optimization, 2008, 4 (2) : 271-285. doi: 10.3934/jimo.2008.4.271 [16] François Monard. Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms. Inverse Problems & Imaging, 2018, 12 (2) : 433-460. doi: 10.3934/ipi.2018019 [17] Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems & Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601 [18] Michael K. Ng, Chi-Pan Tam, Fan Wang. Multi-view foreground segmentation via fourth order tensor learning. Inverse Problems & Imaging, 2013, 7 (3) : 885-906. doi: 10.3934/ipi.2013.7.885 [19] Qun Liu, Lihua Fu, Meng Zhang, Wanjuan Zhang. Two-dimensional seismic data reconstruction using patch tensor completion. Inverse Problems & Imaging, 2020, 14 (6) : 985-1000. doi: 10.3934/ipi.2020052 [20] Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79.

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