Higher-degree eigenvalue complementarity problems for tensors

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.Keywords Tensor · Higher-degree cone eigenvalue · Eigenvalue complementarity problem · Polynomial optimization problem · Augmented Lagrangian method · Alternating direction method of multipliersMathematics Subject Classification (2010) 15A18 · 15A69 · 65K15 · 90C30 · 90C33

Strictly semi-positive tensors and the boundedness of tensor complementarity problems

“…The number and computation of Pareto H(Z)-eigenvalue see Ling, He and Qi [14,15], Chen, Yang and Ye [16].…”

A semismooth Newton method for tensor eigenvalue complementarity problem

“…When the nonnegative cones in (1) are replaced by a closed convex cone and its dual cone, TEiCP is called the cone eigenvalue complementarity problem for high-order tensors by Ling, He and Qi [22]. Moreover, as a natural extension of quadratic eigenvalue complementarity problem for matrices, Ling, He and Qi [23] also consider the high-degree eigenvalue complementarity problem for tensors. Some properties of Pareto eigenvalues are further studied in [39].…”

“…Another kind of complementarity problems related to tensors is considered in [5,16,25,35]. On the other hand, some algorithms for computing the solutions of TEiCP have been proposed, such as shifted projected power method [8], scaling-and-projection algorithm [22] and alternating direction method of multipliers [23]. Notice that all these methods are first order algorithms that are based on gradient information.…”

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.…”

“…Inspired by quadratic eigenvalue complementarity problem (QEiCP) proposed in [32]. Tensor higher-degree eigenvalue complementarity problems (THDEiCP) beyond the framework of TGEiCP and QEiCP is also proposed in [22]. THDEiCP(A, B, C) is to find (λ, x) ∈ R × R n \ {0} such that…”