Further study on tensor absolute value equations

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.

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Acknowledgements

The first author and the third author were supported by National Natural Science Foundation of China (Grant Nos. 11571087 and 11771113) and Natural Science Foundation of Zhejiang Province (Grant No. LY17A010028). The fourth author was supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 15302114, 15300715, 15301716 and 15300717).

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Correspondence to Hongjin He.

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Ling, C., Yan, W., He, H. et al. Further study on tensor absolute value equations. Sci. China Math. 63, 2137–2156 (2020). https://doi.org/10.1007/s11425-018-9560-3

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Keywords

  • tensor absolute value equations
  • H+-tensor
  • P-tensor
  • copositive tensor
  • generalized Newton method

MSC(2010)

  • 15A48
  • 15A69
  • 65K05
  • 90C30
  • 90C20