Higher-degree eigenvalue complementarity problems for tensors

TL;DR

This paper introduces a unified framework for tensor higher-degree eigenvalue complementarity problems, extending beyond matrix quadratic cases, with theoretical analysis and an implementable algorithm for solutions.

Contribution

It develops a comprehensive framework for THDEiCP, including existence results and a practical algorithm, advancing tensor eigenvalue complementarity problem research.

Findings

Established topological properties of higher-degree tensor eigenvalues.

Reformulated THDEiCP as a polynomial optimization problem under symmetry.

Proposed an implementable algorithm with computational results.

Abstract

In this paper, we introduce a unified framework of Tensor Higher-Degree Eigenvalue Complementarity Problem (THDEiCP), which goes beyond the framework of the typical Quadratic Eigenvalue Complementarity Problem (QEiCP) for matrices. First, we study some topological properties of higher-degree cone eigenvalues of tensors. Based upon the symmetry assumptions on the underlying tensors, we then reformulate THDEiCP as a weakly coupled homogeneous polynomial optimization problem, which might be greatly helpful for designing implementable algorithms to solve the problem under consideration numerically. As more general theoretical results, we present the results concerning existence of solutions of THDEiCP without symmetry conditions. Finally, we propose an easily implementable algorithm to solve THDEiCP, and report some computational results.