高阶张量特征值和相关多项式优化问题研究（英文）

Contents
Chapter 1 Introduction 1
11 Eigenvalues problems of higher order tensors 1
12 Related polynomial optimization problems 4
13 Applications 6
14 Spectral properties and algorithms: a literature review 9
15 The organization of this book 14
Chapter 2 Spectral Properties of H-eigenvalue Problems of a Nonnegative Square Tensor 17
21 Introduction 17
22 From nonnegative matrices to nonnegative tensors 18
23 Nonnegative irreducible tensors and primitive tensors 19
24 Perron-Frobenius theorem for nonnegative tensors and related results 22
25 Geometric simplicity 29
26 The Collatz-Wielandt formula 36
27 Other related results 41
28 Some properties for nonnegative weakly irreducible tensors 46
281 Weak irreducibility 46
282 Generalization from nonnegative irreducible tensors to nonnegative
weakly irreducible tensors 50
Chapter 3 Algorithms for Finding the Largest H-eigenvalue of a
Nonnegative Square Tensor 54
31 Introduction 54
32 A polynomial-time approach for computing the spectral radius 55
33 Two algorithms and convergence analysis 57
331 An inexact power-type algorithm 58
332 A one-step inner iteration power-type algorithm 63
34 Numerical experiments 65
341 Experiments on the polynomial-time approach 65
342 Experiments on the inexact algorithms 66
Chapter 4 Spectral Properties and Algorithms of H-singular Value Problems of a Nonnegative Rectangular Tensor 70
41 Introduction 70
42 Preliminaries 70
43 Some conclusions concerning the singular value of a nonnegative
rectangular tensor 72
44 Primitivity and the convergence of the CQZ method for ˉnding the
largest singular value of a nonnegative rectangular tensor 81
45 Algorithms for computing the largest singular value of a nonnegative
rectangular tensor 84
451 A polynomial-time algorithm 84
452 An inexact algorithm 85
46 A solving method of the largest singular value based on the symmetric
embedding 87
461 Singular values of a rectangular tensor 87
462 Singular values of a general tensor 89
Chapter 5 Properties and Algorithms of Z-eigenvalue Problems of a Symmetric Tensor 94
51 Introduction 94
52 Some spectral properties 95
521 The Collatz-Wielandt formula 95
522 Bounds on the Z-spectral radius 99
53 The reformulation problem and the no duality gap result 100
531 The reformulation problem 100
532 Dual problem of (RP) 102
533 No duality gap result 104
54 Relaxations and algorithms 106
541 Nuclear norm regularized convex relaxation of (RP) and the proximal
augmented Lagrangian method 106
542 The truncated nuclear norm regularization and the approximation 112
543 Alternating least eigenvalue method for ˉnding a global minima 114
55 Numerical results 119
Chapter 6 Solving Biquadratic Optimization Problems via
Semideˉnite Relaxation 126
61 Introduction 126
62 Semideˉnite relaxations and approximate bounds 127
621 The nonnegative case 127
622 The square-free case and the positive semideˉnite case 130
63 Approximation al