Structured Rectangular Tensors and Rectangular Tensor Complementarity Problems

Abstract

In this paper, some properties of structured rectangular tensors are presented, and the relationship among these structured rectangular tensors is also given. It is shown that all the V-singular values of rectangular P-tensors are positive. Some necessary and/or sufficient conditions for a rectangular tensor to be a rectangular P-tensor are also obtained. A new subclass of rectangular tensors, which is called rectangular S-tensors, is introduced and it is proved that rectangular S-tensors can be defined by the feasible vectors of the corresponding rectangular tensor complementarity problem.

Rectangular M-tensors and strong rectangular M-tensors

“…Then, a strong rectangular M-tensor is a rectangular S-tensor [17]. From Theorem 11 in [17], the following conclusion can be obtained easily.…”

“…Let = (a i 1 i 2 ...i p j 1 j 2 ... j q ) ∈ [p;q;m;n] , q m ∈ m and q n ∈ n . The rectangular tensor complementarity problem [17], denoted by RTCP( , q m , q n ), is to find vectors x ∈ m and y ∈ n such that q m + x p−1 y q 0, x 0, x T (q m + x p−1 y q ) = 0, q n + x p y q−1 0, y 0, y T (q n + x p y q−1 ) = 0.…”