高阶张量特征值和相关多项式优化问题

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

……

Chapter 7 Approximation Algorithms for Trilinear Optimization with Nonconvex Constraints and Extensions

Chapter 8 Conclusions

References

Index

《数学专著系列丛书：高阶张量特征值和相关多项式优化问题研究（英文）》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。