arXiv subjects

Zhongming Chen

Publications and source records attributed to Zhongming Chen.

23 records · Page 2Linked to original sources

arXiv2016

Further results on $B$-tensors with application to the location of real eigenvalues

In this paper, we give a further study on $B$-tensors and introduce doubly $B$-tensors that contain $B$-tensors. We show that they have similar properties, including their decompositions and strong relationship with strictly (doubly) diagonally dominated tensors. As an application, the properties of $B$-tensors are used to localize real eigenvalues of some tensors, which would be very useful in verifying the positive semi-definiteness of a tensor.

math.SP↗

arXiv2015

Generalized Eigenvalue Complementarity Problem for Tensors

In this paper, the generalized eigenvalue complementarity problem for tensors (GEiCP-T) is addressed, which arises from the stability analysis of finite dimensional mechanical systems and find applications in differential dynamical systems. The general properties of the (GEiCP-T) have been studied. We establish its relationship with the generalized tensor eigenvalue problem. It follows that if exist, the number of $λ$-solutions can be bounded. We also give some sufficient conditions for the existence of the solution. In particular, there exists a unique solution of the (EiCP-T) (i.e., $J=[n]$) for irreducible nonnegative tensors. For the symmetric case, we derive a sufficient and necessary condition for the solvability of the (GEiCP-T) by reformulating it as a nonlinear program. It has also been proved that deciding the solvability of the (EiCP-T) is NP-hard in general. Moreover, a shifted projected power method is proposed to solve the symmetric (GEiCP-T). The monotonic convergence is also established. The numerical experiments demonstrate convergence behavior of our method and show that the algorithm presented is promising.

math.SP↗

arXiv2015

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.

math.OC↗

arXiv2015

The positive semi-definite cone and sum-of-squares cone of Hankel form

In this paper, the geometry properties of Hankel form are studied, including their positive semi-definite (PSD) cone and sum-of-squares (SOS) cone. We denote them by $HPSD(m,n)$ and $HSOS(m,n)$, respectively. We show that both $HPSD(m,n)$ and $HSOS(m,n)$ are closed convex cones. The dual cone of $HPSD(m,n)$ is the convex hull of all $m$-times convolutions of real vectors. Besides, we derive the dual cone of SOS tensors. By reformulation, it follows that the dual cone of $HSOS(m,n)$ can also be written explicitly. These results may lead further research on the Hilbert-Hankel problem.

math.SP↗

arXiv2014

Circulant Tensors with Applications to Spectral Hypergraph Theory and Stochastic Process

Circulant tensors naturally arise from stochastic process and spectral hypergraph theory. The joint moments of stochastic processes are symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of circulant hypergraphs are also symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of directed circulant hypergraphs are circulant tensors, but they are not symmetric in general. In this paper, we study spectral properties of circulant tensors and their applications in spectral hypergraph theory and stochastic process. We show that in certain cases, the largest H-eigenvalue of a circulant tensor can be explicitly identified. In particular, the largest H-eigenvalue of a nonnegative circulant tensor can be explicitly identified. This confirms the results in circulant hypergraphs and directed circulant hypergraphs. We prove that an even order circulant B$_0$ tensor is always positive semi-definite. This shows that the Laplacian tensor and the signless Laplacian tensor of a directed circulant even-uniform hypergraph are positive semi-definite. If a stochastic process is $m$th order stationary, where $m$ is even, then its $m$th order moment, which is a circulant tensor, must be positive semi-definite. In this paper, we give various conditions for a circulant tensor to be positive semi-definite.

math.SP↗