Chan, Chi Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 123-125). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 63-64). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 63-64). Also available on the Internet.

For an inequality system, an error bound is an estimation for the distance from any point to the solution set of the inequality. The Ekeland variational principle (EVP) is an important tool in the study of error bounds. We prove that EVP is equivalent to an error bound result and present several sufficient conditions for an inequality system to have error bounds. In a metric space, a condition is similar to that of Takahashi. In a Banach space we express conditions in terms of an abstract subdifferential and the lower Dini derivative. We then discuss error bounds with exponents by a relation between the lower Dini derivatives of a function and its power function. For an l.s.c. convex function on a reflexive Banach space these conditions turn out to be equivalent. Furthermore a global error bound closely relates to the metric regularity. / Graduate

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Vita. / Bibliography: leaves 135-137. / by Kenneth Sidney Alexander. / Ph.D.

We prove that the blowup term of a blowup solution of Jang's equation on an initial data set (M,g,k) near an arbitrary strictly stable MOTS Σ is exactly −1/√λlog τ, where τ is the distance from Σ and λ is the principal eigenvalue of the MOTS stability operator of Σ. We also prove that the gradient of the solution is of order τ^(-1). Moreover, we apply these results to get a Penrose-like inequality under additional assumptions.

Li Guo-Yin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 105-107). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Examples for the variational inequality problem --- p.2 / Chapter 1.2 --- Approaches for variational inequality problem --- p.7 / Chapter 1.3 --- Error bounds results for variational inequality problem --- p.8 / Chapter 1.4 --- Organization --- p.9 / Chapter 2 --- Solution Theory --- p.11 / Chapter 2.1 --- "Elementary Convex Analysis, Nonsmooth Analysis and Degree theory" --- p.11 / Chapter 2.1.1 --- Elementary Convex Analysis --- p.11 / Chapter 2.1.2 --- Elementary Nonsmooth Analysis --- p.16 / Chapter 2.1.3 --- Degree Theory --- p.18 / Chapter 2.2 --- Existence and Uniqueness Theory --- p.24 / Chapter 3 --- Merit Functions for variational inequalities problem --- p.36 / Chapter 3.1 --- Regularized gap function --- p.38 / Chapter 3.2 --- D-gap function --- p.44 / Chapter 3.3 --- Generalized Regularize gap function and Generalized D-gap function --- p.61 / Chapter 4 --- Error bound results for the merit functions --- p.74 / Chapter 4.1 --- Error bound results for Regularized gap function --- p.77 / Chapter 4.2 --- Error bound results for D-gap function --- p.78 / Chapter 4.3 --- Error bound results for Generalized Regularized gap function --- p.92 / Chapter 4.4 --- Error bound results for Generalized D-gap function --- p.93 / Bibliography --- p.105

Zheng Xi Yin. / "December 2002." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (p. 117-121). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

During the past two decades, numerous recurrent neural networks (RNNs) have been proposed for solving VIs and related problems. However, first, the theories of many emerging RNNs have not been well founded yet; and their capabilities have been underestimated. Second, these RNNs have limitations in handling some types of problems. Third, it is certainly not true that these RNNs are best choices for solving all problems, and new network models with more favorable characteristics could be devised for solving specific problems. / In the research, the above issues are extensively explored from dynamic system perspective, which leads to the following major contributions. On one hand, many new capabilities of some existing RNNs have been revealed for solving VIs and related problems. On the other hand, several new RNNs have been invented for solving some types of these problems. The contributions are established on the following facts. First, two existing RNNs, called TLPNN and PNN, are found to be capable of solving pseudomonotone VIs and related problems with simple bound constraints. Second, many more stability results are revealed for an existing RNN, called GPNN, for solving GVIs with simple bound constraints, and it is then extended to solve linear VIs (LVIs) and generalized linear VIs (GLVIs) with polyhedron constraints. Third, a new RNN, called IDNN, is proposed for solving a special class of quadratic programming problems which features lower structural complexity compared with existing RNNs. Fourth, some local convergence results of an existing RNN, called EPNN, for nonconvex optimization are obtained, and two variants of the network by incorporating two augmented Lagrangian function techniques are proposed for seeking Karush-Kuhn-Tucker (KKT) points, especially local optima, of the problems. / Variational inequality (VI) can be viewed as a natural framework for unifying the treatment of equilibrium problems, and hence has applications across many disciplines. In addition, many typical problems are closely related to VI, including general VI (GVI), complementarity problem (CP), generalized CP (GCP) and optimization problem (OP). / Hu, Xiaolin. / "July 2007." / Adviser: Jun Wang. / Source: Dissertation Abstracts International, Volume: 69-02, Section: B, page: 1102. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 193-207). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract in English and Chinese. / School code: 1307.

by Yang Weihong. / "March 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 89-92). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.