On the cone eigenvalue complementarity problem for higher-order tensors

TL;DR

This paper introduces the tensor generalized eigenvalue complementarity problem (TGEiCP), proves its solvability, develops optimization reformulations, establishes bounds on eigenvalues, and proposes a projection algorithm with preliminary computational results.

Contribution

It is the first to analyze TGEiCP's solvability, provide optimization reformulations, and develop an implementable solution algorithm.

Findings

TGEiCP is solvable under certain conditions.

Upper bounds for cone eigenvalues of tensors are established.

A projection algorithm for solving TGEiCP is developed and tested.

Abstract

In this paper, we consider the tensor generalized eigenvalue complementarity problem (TGEiCP), which is an interesting generalization of matrix eigenvalue complementarity problem (EiCP). First, we given an affirmative result showing that TGEiCP is solvable and has at least one solution under some reasonable assumptions. Then, we introduce two optimization reformulations of TGEiCP, thereby beneficially establishing an upper bound of cone eigenvalues of tensors. Moreover, some new results concerning the bounds of number of eigenvalues of TGEiCP further enrich the theory of TGEiCP. Last but not least, an implementable projection algorithm for solving TGEiCP is also developed for the problem under consideration. As an illustration of our theoretical results, preliminary computational results are reported.