Positive Definite Tensors to Nonlinear Complementarity Problems

The main purpose of this note is to investigate some kinds of nonlinear complementarity problems (NCP). For the structured tensors, such as, symmetric positive definite tensors and copositive tensors, we derive the existence theorems on a solution of these kinds of nonlinear complementarity problems. We prove that a unique solution of the NCP exists under the condition of diagonalizable tensors.Comment: 11 page

Properties of Some Classes of Structured Tensors

In this paper, we extend some classes of structured matrices to higher order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. The potential links of such structured tensors with optimization, nonlinear equations, nonlinear complementarity problems, variational inequalities and the nonnegative tensor theory are also discussed.Comment: arXiv admin note: text overlap with arXiv:1405.1288 by other author

A Semismooth Newton Method for Tensor Eigenvalue Complementarity Problem

In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By introducing an NCP-function, we reformulate the tensor eigenvalue complementarity problem as a system of nonlinear equations. We show that this function is strongly semismooth but not differentiable, in which case the classical smoothing methods cannot apply. Furthermore, we propose a damped semismooth Newton method for tensor eigenvalue complementarity problem. A new procedure to evaluate an element of the generalized Jocobian is given, which turns out to be an element of the B-subdifferential under mild assumptions. As a result, the convergence of the damped semismooth Newton method is guaranteed by existing results. The numerical experiments also show that our method is efficient and promising

Tensor Complementarity Problem and Semi-positive Tensors

The tensor complementarity problem (\q, \mathcal{A}) is to \mbox{ find } \x \in \mathbb{R}^n\mbox{ such that }\x \geq \0, \q + \mathcal{A}\x^{m-1} \geq \0, \mbox{ and }\x^\top (\q + \mathcal{A}\x^{m-1}) = 0. We prove that a real tensor $\mathcal{A}$ is a (strictly) semi-positive tensor if and only if the tensor complementarity problem (\q, \mathcal{A}) has a unique solution for \q>\0 (\q\geq\0), and a symmetric real tensor is a (strictly) semi-positive tensor if and only if it is (strictly) copositive. That is, for a strictly copositive symmetric tensor $\mathcal{A}$, the tensor complementarity problem (\q, \mathcal{A}) has a solution for all \q \in \mathbb{R}^n