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


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Author: Yang Yuning
Language: English
ISBN/ISSN: 9787030437655
2015-11;  Soft Cover

Chapter 1 Introduction
1.1 Eigenvalues problems of higher order tensors
1.2 Related polynomial optimization problems
1.3 Applications
1.4 Spectral properties and algorithms: a literature review
1.5 The organization of this book
Chapter 2 Spectral Properties of H-eigenvalue Problems of a Nonnegative Square Tensor
2.1 Introduction
2.2 From nonnegative matrices to nonnegative tensors
2.3 Nonnegative irreducible tensors and primitive tensors
2.4 Perron-Frobenius theorem for nonnegative tensors and related results
2.5 Geometric simplicity
2.6 The Collatz-Wielandt formula
2.7 Other related results
2.8 Some properties for nonnegative weakly irreducible tensors
2.8.1 Weak irreducibility
2.8.2 Generalization from nonnegative irreducible tensors to nonnegative weakly irreducible tensors
Chapter 3 Algorithms for Finding the Largest H-eigenvalue of a Nonnegative Square Tensor
3.1 Introduction
3.2 A polynomial-time approach for computing the spectral radius
3.3 Two algorithms and convergence analysis
3.3.1 An inexact power-type algorithm
3.3.2 A one-step inner iteration power-type algorithm
3.4 Numerical experiments
3.4.1 Experiments on the polynomial-time approach
3.4.2 Experiments on the inexact algorithms
Chapter 4 Spectral Properties and Algorithms of H-singular Value Problems of a Nonnegative Rectangular Tensor
4.1 Introduction
4.2 Preliminaries
4.3 Some conclusions concerning the singular value of a nonnegative rectangular tensor
4.4 Primitivity and the convergence of the CQZ method for fnding the largest singular value of a nonnegative rectangular tensor
4.5 Algorithms for computing the largest singular value of a nonnegative rectangular tensor
4.5.1 A polynomial-time algorithm
4.5.2 An inexact algorithm
4.6 A solving method of the largest singular value based on the symmetric embedding
4.6.1 Singular values of a rectangular tensor
4.6.2 Singular values of a general tensor
Chapter 5 Properties and Algorithms of Z-eigenvalue Problems of a Symmetric Tensor
5.1 Introduction
5.2 Some spectral properties
5.2.1 The Collatz-Wielandt formula
5.2.2 Bounds on the Z-spectral radius
5.3 The reformulation problem and the no duality gap result
5.3.1 The reformulation problem
5.3.2 Dual problem of (RP)
5.3.3 No duality gap result
5.4 Relaxations and algorithms
5.4.1 Nuclear norm regularized convex relaxation of (RP) and the proximal augmented Lagrangian method
5.4.2 The truncated nuclear norm regularization and the approximation
5.4.3 Alternating least eigenvalue method for fnding a global minima
5.5 Numerical results

Chapter 6 Solving Biquadratic Optimization Problems via Semidefnite Relaxation
6.1 Introduction
6.2 Semidefnite relaxations and approximate bounds
6.2.1 The nonnegative case
6.2.2 The square-free case and the positive semidefnite case
6.3 Approximation algorithms for Trilinear Optimization with Nonconvex Constraints and Extensions
6.3.1 Approximation method for the nonnegative case
6.3.2 The binary biquadratic optimization problem
6.3.3 A generalization of the binary biquadratic optimization
6.4 Numerical experiments
Chapter 7 Approximation Algorithms for Trilinear Optimization with Nonconvex Constraints and Extensions
7.1 Introduction
7.2 A powerful approach to solve the trilinear optimization problem over unit spheres
7.3 Quadratic constraints
7.4 A special case
7.5 Extending to the biquadratic case

Chapter 8 Conclusions

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