NACO
Some results on $l^k$-eigenvalues of tensor and related spectral radius
Chen Ling Liqun Qi
Numerical Algebra, Control & Optimization 2011, 1(3): 381-388 doi: 10.3934/naco.2011.1.381
In this paper, we study the $l^k$-eigenvalues/vectors of a real symmetric square tensor. Specially, we investigate some properties on the related $l^k$-spectral radius of a real nonnegative symmetric square tensor.
keywords: $l^k$-eigenvalue irreducibility. $l^k$-spectral radius Nonnegative tensor
JIMO
Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors
Haibin Chen Liqun Qi
Journal of Industrial & Management Optimization 2015, 11(4): 1263-1274 doi: 10.3934/jimo.2015.11.1263
Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only if its generating vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its generating vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semi-definiteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised.
keywords: generating vector. Cauchy tensor eigenvalue positive definiteness Positive semi-definiteness
JIMO
Test of copositive tensors
Li Li Xinzhen Zhang Zheng-Hai Huang Liqun Qi
Journal of Industrial & Management Optimization 2018, 13(5): 1-11 doi: 10.3934/jimo.2018075

In this paper, an SDP relaxation algorithm is proposed to test the copositivity of higher order tensors. By solving finitely many SDP relaxations, the proposed algorithm can determine the copositivity of higher order tensors. Furthermore, for any copositive but not strictly copositive tensor, the algorithm can also check it exactly. Some numerical results are reported to show the efficiency of the proposed algorithm.

keywords: Symmetric tensor polynomial optimization SDP relaxation
JIMO
Circulant tensors with applications to spectral hypergraph theory and stochastic process
Zhongming Chen Liqun Qi
Journal of Industrial & Management Optimization 2016, 12(4): 1227-1247 doi: 10.3934/jimo.2016.12.1227
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 an even order circulant tensor to be positive semi-definite.
keywords: eigenvalues of tensors Circulant tensors directed circulant hypergraphs circulant hypergraphs positive semi-definiteness.
JIMO
Semismooth reformulation and Newton's method for the security region problem of power systems
Liqun Qi Zheng yan Hongxia Yin
Journal of Industrial & Management Optimization 2008, 4(1): 143-153 doi: 10.3934/jimo.2008.4.143
In this paper, we investigate the reformulation of the steady state security region problem for electrical power systems. Firstly, a simple security region problem with one changeable parameter is reformulated into a system of semismooth equations, which is composed by the normal power flow equations and an additional piecewise smooth equation. Then the semismooth Newton method and the smoothing Newton method can be applied to solve the problem. Preliminary numerical results show that the method is promising. Finally, by using the smoothing technique, a more complicated security region problem, the Euclidean security region problem, is reformulated as an equality constrained optimization problem. These works provide a possibility to implement on-line calculation of the security region of electrical power systems.
keywords: Power system security region local superlinear convergence. global convergence smoothing technique semismooth Newton method
JIMO
A network simplex algorithm for simple manufacturing network model
Jiangtao Mo Liqun Qi Zengxin Wei
Journal of Industrial & Management Optimization 2005, 1(2): 251-273 doi: 10.3934/jimo.2005.1.251
In this paper, we propose a network model called simple manufacturing network. Our model is a combined version of the ordinary multicommodity network and the manufacturing network flow model. It can be used to characterize the complicated manufacturing scenarios. By formulating the model as a minimum cost flow problem plus several bounded variables, we present a modified network simplex method, which exploits the special structure of the model and can perform the computation on the network. A numerical example is provided for illustrating our method.
keywords: distilling process. network simplex algorithm Multicommodity network minimum cost flow optimization model

Year of publication

Related Authors

Related Keywords

[Back to Top]