Tensor Complementarity Problems—Part I: Basic Theory

A note on the solvability of a tensor equation

“…More importantly, in [1,3], it is shown that if the leading tensor A k is nonsingular, then the system (2) definitely has a solution. However, it would be restrictive on posing the nonsingularity of the leading tensor solely in applications [4]. In this note, we further generalize the results in [1,3] and present a theorem for possible singular A k .…”

Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems

“…For instance, various tensors with special structures were given in [13,29,30,33,46], including copositive tensors, M tensors, P -tensors and positive-definite tensors. On the other hand, many kinds of tensor optimization problem have been proposed, such as tensor complementarity problems (TCP) in [3,4,14,15,17,18,31,35,36,38,39,47,50], tensor eigenvalue problems (TEiP) in [7,19,25,41,43] and tensor eigenvalue complementarity problems (TEiCP) in [9,10,16,21,22,44]. As an important special case of complementarity problems, tensor eigenvalue complementarity problems have been developing rapidly since the past decades.…”

Weakening convergence conditions of a potential reduction method for tensor complementarity problems

“…Introduction. As a generalization of the linear complementarity problem [4], the tensor complementarity problem (TCP for short) has been introduced and investigated in [3,11,22], which is a specific class of nonlinear complementarity problems [6]. The TCP has many applications, including n-person non-cooperative games [10], hypergraph clustering [12], a class of traffic equilibrium problems [12], and so on.…”

“…Che, Qi and Wei [3] established the non-emptiness and compactness of the solution set for the TCP with the underlying tensor being positive definite; furthermore, they showed that the tensor complementarity problem with a diagonalizable and positive definite tensor has a unique solution. Theoretical advances in tensor complementarity problems are summarized in [11].…”

“…From Definition 2.1, it is easy to see that each strictly positive definite tensor must be a positive definite tensor, a strictly positive definite tensor is a strong P tensor , and a strong P tensor is a P tensor. In addition, a positive definite tensor is a P tensor [19], a P tensor is an R tensor [11]. Furthermore, a positive definite tensor is an R tensor, and is also an R 0 tensor.…”