A Study on Eigencalues of Higher-Order Tensors and Related Polynomial Op timization Problems

高阶张量特征值和相关多项式优化问题研究(英文)

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Author: Qingzhi-Yang & Yuning-Yang
Language: English
ISBN/ISSN: 9787030437655
Published on: 2015-03
Paperback

本书主要研究张量特征值的各种性质、最大特征值的计算方法及相关的的的的计算。具体内容包括:正方张量特征值的定义及基本性质、非负正方张量的Perron-Frobenius定理,张量不可约的判定条件、不可约非负张量H特征值的分布、H特征值的几何单性,基于Perron-Frobenius定理表明一定条件下谱半径可以在多项式时间内求解等。



Chapter 1 Introduction
1.1Eigenvalues problems of higher order tensors
1.2Related polynomial optimization problems
1.3Applications
1.4Spectral properties and algorithms: a literature review
1.5The organization of this book
Chapter 2 Spectral Properties of H-eigenvalue Problems of a Nonnegative Square Tensor
2.1Introduction
2.2From nonnegative matrices to nonnegative tensors
2.3Nonnegative irreducible tensors and primitive tensors
2.4Perron-Frobenius theorem for nonnegative tensors and related results
2.5Geometric simplicity
2.6The Collatz-Wielandt formula
2.7Other related results
2.8Some properties for nonnegative weakly irreducible tensors
2.8.1Weak irreducibility
2.8.2Generalization from nonnegative irreducible tensors to nonnegative weakly irreducible tensors
Chapter 3 Algorithms for Finding the Largest H-eigenvalue of a Nonnegative Square Tensor
3.1Introduction
3.2A polynomial-time approach for computing the spectral radius
3.3Two algorithms and convergence analysis
3.3.1An inexact power-type algorithm
3.3.2A one-step inner iteration power-type algorithm
3.4Numerical experiments
3.4.1Experiments on the polynomial-time approach
3.4.2Experiments on the inexact algorithms
Chapter 4 Spectral Properties and Algorithms of H-singular Value Problems of a Nonnegative Rectangular Tensor
4.1Introduction
4.2Preliminaries
4.3Some conclusions concerning the singular value of a nonnegative rectangular tensor
4.4Primitivity and the convergence of the CQZ method for fnding the largest singular value of a nonnegative rectangular tensor
4.5Algorithms for computing the largest singular value of a nonnegative rectangular tensor
4.5.1A polynomial-time algorithm
4.5.2An inexact algorithm
4.6A solving method of the largest singular value based on the symmetric embedding
4.6.1Singular values of a rectangular tensor
4.6.2Singular values of a general tensor
Chapter 5 Properties and Algorithms of Z-eigenvalue Problems of a Symmetric Tensor
5.1Introduction
5.2Some spectral properties
5.2.1The Collatz-Wielandt formula
5.2.2Bounds on the Z-spectral radius
5.3The reformulation problem and the no duality gap result
5.3.1The reformulation problem
5.3.2Dual problem of (RP)
5.3.3No duality gap result
5.4Relaxations and algorithms
5.4.1Nuclear norm regularized convex relaxation of (RP) and the proximal augmented Lagrangian method
5.4.2The truncated nuclear norm regularization and the approximation
5.4.3Alternating least eigenvalue method for fnding a global minima
5.5Numerical results
Chapter 6 Solving Biquadratic Optimization Problems via Semidefnite Relaxation
6.1Introduction
6.2Semidefnite relaxations and approximate bounds
6.2.1The nonnegative case
6.2.2The square-free case and the positive semidefnite case
6.3Approximation algorithms for Trilinear Optimization with Nonconvex Constraints and Extensions
6.3.1Approximation method for the nonnegative case
6.3.2The binary biquadratic optimization problem
6.3.3A generalization of the binary biquadratic optimization
6.4Numerical experiments
Chapter 7Approximation Algorithms for Trilinear Optimization with Nonconvex Constraints and Extensions
7.1Introduction
7.2A powerful approach to solve the trilinear optimization problem over unit spheres
7.3Quadratic constraints
7.4A special case
7.5Extending to the biquadratic case
Chapter 8Conclusions
References
Index


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