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A Study on Eigenvalues of Higher-Order Tensors and Related Polynomial Optimization Problems (高阶张量特征值和相关多项式优化问题研究)


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A Study on Eigenvalues of Higher-Order Tensors and Related Polynomial Optimization Problems (高阶张量特征值和相关多项式优化问题研究)
  • 书号：9787030437655
    作者：
  • 外文书名：
  • 装帧：平装
    开本：B5
  • 页数：196
    字数：300
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  • 出版社：
    出版时间：2015-12-03
  • 所属分类：
  • 定价： ￥78.00元
    售价： ￥61.62元
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目录

  • 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 algorithms for the biquadratic optimization problems 133
    631 Approximation method for the nonnegative case 133
    632 The binary biquadratic optimization problem 136
    633 A generalization of the binary biquadratic optimization 141
    64 Numerical experiments 143
    Chapter 7 Approximation Algorithms for Trilinear Optimization
    with Nonconvex Constraints and Extensions 146
    71 Introduction 146
    72 A powerful approach to solve the trilinear optimization problem over
    unit spheres 149
    73 Quadratic constraints 154
    74 A special case 161
    75 Extending to the biquadratic case 167
    Chapter 8 Conclusions 169
    References 170
    Index 184
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