J Optim Theory Appl (2018) 176:289–305
Structural Properties of Tensors and Complementarity
· Wei Mei
Received: 5 July 2017 / Accepted: 18 December 2017 / Published online: 2 January 2018
© Springer Science+Business Media, LLC, part of Springer Nature 2017
Abstract In this paper, one of our main purposes is to prove the boundedness of the
solution set of tensor complementarity problems such that the speciﬁc bounds depend
only on the structural properties of such a tensor. To achieve this purpose, ﬁrstly,
we prove that this class of structured tensors is strictly semi-positive. Subsequently,
the strictly lower and upper bounds of operator norms are given for two positively
homogeneous operators. Finally, with the help of the above upper bounds, we show
that the solution set of tensor complementarity problems has the strictly lower bound.
Furthermore, the upper bounds of spectral radius are obtained, which depends only
on the principal diagonal entries of tensors.
Keywords Structured tensor · Tensor complementarity problems · Spectral radius ·
Operator norms · Upper and lower bounds
Mathematics Subject Classiﬁcation 47H15 · 47H12 · 34B10 · 47A52 · 47J10 ·
47H09 · 15A48 · 47H07
As a natural extension of linear complementarity problem, the tensor complemen-
tarity problem is a new topic emerged from the tensor community. Meanwhile, such
Communicated by Liqun Qi.
School of Mathematics and Information Science, Henan Normal University,
XinXiang, HeNan, People’s Republic of China