Science China Mathematics

, Volume 61, Issue 9, pp 1695–1710 | Cite as

Tensor absolute value equations

  • Shouqiang Du
  • Liping Zhang
  • Chiyu Chen
  • Liqun Qi


This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.


M-tensors absolute value equations Levenberg-Marquardt method tensor complementarity problem 


15A48 15A69 65K05 90C30 90C20 


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This work was supported by National Natural Science Foundation of China (Grant Nos. 11671220, 11401331, 11771244 and 11271221), the Nature Science Foundation of Shandong Province (Grant Nos. ZR2015AQ013 and ZR2016AM29), and the Hong Kong Research Grant Council (Grant Nos. PolyU 501913, 15302114, 15300715 and 15301716). The authors thank the anonymous referees for their constructive comments and suggestions which led to a significantly improved version of the paper.


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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Shouqiang Du
    • 1
  • Liping Zhang
    • 2
  • Chiyu Chen
    • 2
  • Liqun Qi
    • 3
  1. 1.School of Mathematics and StatisticsQingdao UniversityQingdaoChina
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  3. 3.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHong KongChina

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