PolyU IR Collection: AMA Theses
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Some nonlinear spectral properties of higher order tensors
http://hdl.handle.net/10397/6495
Title: Some nonlinear spectral properties of higher order tensors<br/><br/>Authors: Song, Yisheng<br/><br/>Abstract: The main purposes of this thesis focus on the nonlinear spectral properties of higher order tensor with the help of the spectral theory and fixed point theory of nonlinear positively homogeneous operator as well as the constrained minimization theory of homogeneous polynomial. The main contributions of this thesis are as follows. We obtain the Fredholm alternative theorems of the eigenvalue (included E-eigenvalue, H-eigenvalue, Z-eigenvalue) of a higher order tensor A. Some relationship between the Gelfand formula and the spectral radius are discussed for the spectra induced by such several classes of eigenvalues of a higher order tensor. This content is mainly based on the paper 5 in Underlying Papers. We show that the eigenvalue problem of a nonnegative tensor A can be viewed as the fixed point problem of the Edelstein Contraction with respect to Hilbert's projective metric. Then by means of the Edelstein Contraction Theorem, we deal with the existence and uniqueness of the positive eigenvalue-eigenvector of such a tensor, and give an iteration sequence for finding positive eigenvalue of such a tensor, i.e., a nonlinear version of the famous Krein-Rutman Theorem. This content is mainly based on the paper 2 in Underlying Papers. We introduce the concept of eigenvalue to the additively homogeneous mapping pairs (f, g), and establish existence and uniqueness of such a eigenvalue under the boundedness of some orbits of f, g in the Hilbert semi-norm. In particular, the nonlinear Perron-Frobenius property for nonnegative tensor pairs (A, B) is given without involving the calculation of the tensor inversion. Moreover, we also present the iteration methods for finding generalized H-eigenvalue of nonnegative tensor pairs (A, B). This content is mainly based on the paper 1 in Underlying Papers. We introduce the concepts of Pareto H-eigenvalue and Pareto Z-eigenvalue of higher order tensor for studying constrained minimization problem and show the necessary and sufficient conditions of such eigenvalues. We obtain that a symmetric tensor has at least one Pareto H-eigenvalue (Pareto Z-eigenvalue). What is more, the minimum Pareto H-eigenvalue (or Pareto Z-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor A is copositive if and only if every Pareto H-eigenvalue (Z.eigenvalue) of A is non-negative. This content is mainly based on the papers 3 and 4 in Underlying Papers.<br/><br/>Description: 101 p. ; 30 cm.; PolyU Library Call No.: [THS] LG51 .H577P AMA 2013 SongSpectral hypergraph theory
http://hdl.handle.net/10397/6477
Title: Spectral hypergraph theory<br/><br/>Authors: Hu, Shenglong<br/><br/>Abstract: The main subject of this thesis is the study of a few basic problems in spectral hypergraph theory based on Laplacian-type tensors. These problems are hypergraph analogues of some important problems in spectral graph theory. As some foundations, we study some new problems of tensor determinant and non-negative tensor partition. Then two classes of Laplacian-type tensors for uniform hypergraphs are proposed. One is called Laplacian, and the other one Laplace-Beltrami tensor. We study the H-spectra of uniform hypergraphs through their Laplacian, and the Z-spectra of even uniform hypergraphs through their Laplace-Beltrami tensors. All the H{204}-eigenvalues of the Laplacian can be computed out through the developed partition method. Spectral component, an intrinsic notion of a uniform hypergraph, is introduced to characterize the hypergraph spectrum. Many fundamental properties of the spectrum are connected to the underlying hypergraph structures. Basic spectral hypergraph theory based on Laplacian-type tensors are built. With the theory, we study algebraic connectivity, edge connectivity, vertex connectivity, edge expansion, and spectral invariance of the hypergraph.<br/><br/>Description: xii, 107 p. : ill. ; 30 cm.; PolyU Library Call No.: [THS] LG51 .H577P AMA 2013 HuSOptimality conditions of semi-infinite programming and generalized semi-infinite programming
http://hdl.handle.net/10397/6420
Title: Optimality conditions of semi-infinite programming and generalized semi-infinite programming<br/><br/>Authors: Chen, Zhangyou<br/><br/>Abstract: Semi-infinite programming has been long an important model of optimization problems, arising from areas such as approximation, control, probability. Generalized semi-infinite programming has also been an active research area with relatively short history. Nonetheless it has been known that the study of a generalized semi-infinite programming problem is much more difficult than a semi-infinite programming problem. The purpose of this thesis is to develop necessary optimality conditions for semi-infinite and generalized semi-infinite programming problems with penalty functions techniques as well as other approaches. We introduce two types of p-th order penalty functions (0 < p ≤ 1), for semi-infinite programming problems, and explore various relations between them and their relations with corresponding calmness conditions. Under the exactness of certain type penalty functions and some other appropriate conditions especially second order conditions of the constraint functions, we develop optimality conditions for semi-infinite programming problems. This process is also applied to generalized semi-infinite programming problems after being equivalently transformed into standard semi-infinite programming problems. Via the transformation of penalty functions of the lower level problems, we study some properties of the feasible set of the generalized semi-infinite programming problem which is known to possess unusual properties such as non-closedness, re-entrant corners, disjunctive structures, and further establish a sequence of approximate optimization problems and approximate properties for generalized semi-infinite programming problems. We also investigate nonsmooth generalized semi-infinite programming problems via generalization differentiation and derive corresponding optimality conditions via variational analysis tools. Finally, we characterize the strong duality theory of generalized semi-infinite programming problems with convex lower level problems via generalized augmented Lagrangians.<br/><br/>Description: vi, 111 p. ; 30 cm.; PolyU Library Call No.: [THS] LG51 .H577P AMA 2013 ChenAlgorithms and applications of semidefinite space tensor conic convex program
http://hdl.handle.net/10397/6405
Title: Algorithms and applications of semidefinite space tensor conic convex program<br/><br/>Authors: Xu, Yi<br/><br/>Abstract: This thesis focuses on studying the algorithms and applications of positive semi-definite space tensors. A positive semi-definite space tensors are a special type semi-definite tensors with dimension 3. Positive semi-definite space tensors have some applications in real life, such as the medical imaging. However, there isn't an algorithm with good performance to solve an optimization problem with the positive semi-definite space tensor constraint, and the structure of positive semi-definite space tensors is not well explored. In this thesis, firstly, we try to analysis the properties of positive semi-definite space tensors; Then, we construct practicable algorithms to solve an optimization problem with the positive semi-definite space tensor constraint; Finally we use positive semi-definite space tensors to solve some medical problems. The main contributions of this thesis are shown as follows. Firstly, we study the methods to verify the semi-definiteness of space tensors and the properties of H-eigenvalue of tensors. As a basic property of space tensors, the positive semi-definiteness show significant importance in theory. However, there is not a good method to verify the positive semi-definiteness of space tensors. Based upon the nonnegative polynomial theory, we present two methods to verify whether a space tensor positive semi-definite or not. Furthermore, we study the smallest H-eigenvalue of tensors by the relationship between the smallest H-eigenvalue of tensors and their positive semi-definiteness.; Secondly, we consider the positive semi-definite space tensor cone constrained convex program, its structure and algorithms. We study defining functions, defining sequences and polyhedral outer approximations for this positive semi-definite space tensor cone, give an error bound for the polyhedral outer approximation approach, and thus establish convergence of three polyhedral outer approximation algorithms for solving this problem. We then study some other approaches for solving this structured convex program. These include the conic linear programming approach, the nonsmooth convex program approach and the bi-level program approach. Some numerical examples are presented. Thirdly, we apply positive semi-definite tensors into medical brain imagining. Because of the well-known limitations of diffusion tensor imaging (DTI) in regions of low anisotropy and multiple fiber crossing, high angular resolution diffusion imaging (HARDI) and Q-Ball Imaging (QBI) are used to estimate the probability density function (PDF) of the average spin displacement of water molecules. In particular, QBI is used to obtain the diffusion orientation distribution function (ODF) of these multiple fiber crossing. The ODF, as a probability distribution function, should be nonnegative. We propose a novel technique to guarantee nonnegative ODF by minimizing a positive semi-definite space tensor convex optimization problem. Based upon convex analysis and optimization theory, we derive its optimality conditions. And then we propose a gradient descent algorithm for solving this problem. We also present formulas for determining the principal directions (maxima) of the ODF. Numerical examples on synthetic data as well as MRI data are displayed to demonstrate our approach.<br/><br/>Description: xx, 83 p. : ill. ; 30 cm.; PolyU Library Call No.: [THS] LG51 .H577P AMA 2013 Xu