Two aspects of the theory of barotropic modons are examined in this thesis. First, sufficient neutral stability conditions are derived in the form of an integral constraint for westward and eastward-travelling modons. It is shown that eastward-travelling and westward-travelling modons are neutrally stable to perturbations in which the energy is contained mainly in spectral components with wavenumber magnitudes (|ƞ|) satisfying |ƞ|<κ and |ƞ|>κ, respectively, where κ is the modon wavenumber. These results imply that when κ/|ƞ|>1 the slope of the neutral stability curve proposed by McWilliams et al.(l98l) for eastward-travelling modons must begin to increase as κ/|ƞ| increases. The neutral stability condition is computed with mesoscale wavenumber eddy energy spectra representative of the atmosphere and ocean. Eastward-travelling atmospheric modons are neutrally stable to the observed seasonally- and annually-averaged atmospheric eddies. The neutral stability of westward-travelling atmospheric modons and oceanic modons cannot be inferred on the basis of the observed wavenumber eddy energy spectra for the atmosphere and ocean. Second, a leading order perturbation theory is developed to describe the propagation of barotropic modons in a slowly varying medium. Two problems are posed and solved. A perturbation solution is obtained describing the propagation of an eastward-travelling modon modulated by a weak bottom Ekman boundary layer. The results predict that the modon radius and translation speed decay exponentially and that the modon wavenumber increases exponentially, resulting in an exponential amplitude decay in the streamfunction and vorticity. These results agree with the numerical solution of McWilliams et al.(l98l). A leading order perturbation theory is also developed describing modon propagation over slowly varying topography. Nonlinear hyperbolic equations are derived to describe the evolution of the slowly varying modon radius, translation speed and wavenumber for arbitrary finite-amplitude topography. To leading order, the modon is unaffected by meridional gradients in topography. Analytical perturbation solutions for the modon radius, translation speed and wavenumber are obtained for small-amplitude topography. The perturbations take the form of westward and eastward-travelling transients and a stationary component proportional to the topography. The general solution is applied to ridge-like and escarpment-like topographic configurations. / Science, Faculty of / Mathematics, Department of / Graduate

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.

Let T be a multivalued mapping from a nonempty subset of a topological vector space into its topological dual. In this paper, we discuss the relationship between the multivalued mapping T satisfying the (S)_+ condition and T satisfying the (S)_+^1 condition. To unify the (S)_+ condition for single-valued and multivalued mappings, we introduce the weak (S)_+ condition for single-valued mappings defined in [9] to multivalued mappings. The above conditions extend naturally to mappings into L(X,Z), where Z is an ordered Hausdorff topological vector space. We also derive some existence results for generalized vector variational inequalities and generalized variational inequalities associated with mappings which satisfy the (S)_+, (S)_+^1 or weak (S)_+ condition.

Esta dissertação busca aproveitar os métodos de penalidades exatas diferenciáveis de programação não-linear para resolver problemas de desigualdades variacionais. Problemas desse tipo têm recebido grande atenção na literatura recentemente e possuem aplicações em diversas áreas como Engenharia, Física e Economia. Métodos de penalidades exatas diferenciáveis foram desenvolvidos nos anos 70 e 80 para resolver problemas de otimização com restrições por meio da solução de problemas irrestritos. Esses problemas são tais que, com uma escolha apropriada do parâmetro de penalização, uma solução do problema original é recuperada após a resolução de um único problema irrestrito. A função a ser minimizada é semelhante a um lagrangiano aumentado clássico, porém uma estimativa do multiplicador é automaticamente calculada a partir do ponto primal. Nesse trabalho, mostramos como acoplar a estimativa de multiplicadores sugerida por Glad e Polak [27] ao lagrangiano aumentado clássico para desigualdades variacionais sugerido por Auslender e Teboulle. Obtivemos assim uma penalidade exata para problemas de desigualdades variacionais. Os resultados mais finos de exatidão foram obtidos no caso de problemas de complementaridade não-linear. Uma característica importante da penalidade proposta é que ela não envolve informações de segunda ordem das funções que definem a desigualdade variacional. Além desses resultados, que formam o núcleo da dissertação, apresentamos uma breve revisão de penalidades não-exatas diferenciáveis , exatas não-diferenciáveis e exatas diferenciáveis em otimização. / This work intends to build upon differentiable exact penalty methods for nonlinear programming, using them to solve variational inequality problems. Such problems have been given a lot of attention in the literature lately and have applications to diverse areas of knowledge such as Engineering, Physics and Economics. Differentiable exact penalty methods were developed during the 70s and 80s to solve constrained optimization problems by means of the solution of unconstrained problems. Those problems are such that, with an appropriate choice of the penalty parameter, one finds a solution of the original constrained problem by solving only one unconstrained problem. The function which is minimized is similar to the classic augmented lagrangian, but an estimate of the multiplier is automatically calculated from the primal point. In this thesis we show how to couple Glad and Polak?s multiplier estimate, with the classic augmented lagrangian of a variational inequality developed by Auslender and Teboulle. This allowed us to obtain an exact penalty function for variational inequality problems. The best exactness results were obtained in the particular case of nonlinear complementarity problems. An important characteristic of the proposed penalty is that it doesn?t involve second order information of any of the functions which compose the variational inequality. In addition to those results, which are the core of this work, we also present a brief review of inexact differentiable penalties, exact nondifferentiable penalties and differentiable exact penalties in optimization.

Esta dissertação busca aproveitar os métodos de penalidades exatas diferenciáveis de programação não-linear para resolver problemas de desigualdades variacionais. Problemas desse tipo têm recebido grande atenção na literatura recentemente e possuem aplicações em diversas áreas como Engenharia, Física e Economia. Métodos de penalidades exatas diferenciáveis foram desenvolvidos nos anos 70 e 80 para resolver problemas de otimização com restrições por meio da solução de problemas irrestritos. Esses problemas são tais que, com uma escolha apropriada do parâmetro de penalização, uma solução do problema original é recuperada após a resolução de um único problema irrestrito. A função a ser minimizada é semelhante a um lagrangiano aumentado clássico, porém uma estimativa do multiplicador é automaticamente calculada a partir do ponto primal. Nesse trabalho, mostramos como acoplar a estimativa de multiplicadores sugerida por Glad e Polak [27] ao lagrangiano aumentado clássico para desigualdades variacionais sugerido por Auslender e Teboulle. Obtivemos assim uma penalidade exata para problemas de desigualdades variacionais. Os resultados mais finos de exatidão foram obtidos no caso de problemas de complementaridade não-linear. Uma característica importante da penalidade proposta é que ela não envolve informações de segunda ordem das funções que definem a desigualdade variacional. Além desses resultados, que formam o núcleo da dissertação, apresentamos uma breve revisão de penalidades não-exatas diferenciáveis , exatas não-diferenciáveis e exatas diferenciáveis em otimização. / This work intends to build upon differentiable exact penalty methods for nonlinear programming, using them to solve variational inequality problems. Such problems have been given a lot of attention in the literature lately and have applications to diverse areas of knowledge such as Engineering, Physics and Economics. Differentiable exact penalty methods were developed during the 70s and 80s to solve constrained optimization problems by means of the solution of unconstrained problems. Those problems are such that, with an appropriate choice of the penalty parameter, one finds a solution of the original constrained problem by solving only one unconstrained problem. The function which is minimized is similar to the classic augmented lagrangian, but an estimate of the multiplier is automatically calculated from the primal point. In this thesis we show how to couple Glad and Polak?s multiplier estimate, with the classic augmented lagrangian of a variational inequality developed by Auslender and Teboulle. This allowed us to obtain an exact penalty function for variational inequality problems. The best exactness results were obtained in the particular case of nonlinear complementarity problems. An important characteristic of the proposed penalty is that it doesn?t involve second order information of any of the functions which compose the variational inequality. In addition to those results, which are the core of this work, we also present a brief review of inexact differentiable penalties, exact nondifferentiable penalties and differentiable exact penalties in optimization.

First, we study the associated regularized gap functions and the D-gap functions and compute their Clarke-Rockafellar directional derivatives and the Clarke generalized gradients. Second, using these tools and extending the works of Fukushima and Pang (who studied the case when F is smooth), we present results on the relationship between minimizing sequences and stationary sequences of the D-gap functions, regardless the existence of solutions of (VIP). Finally, as another application, we show that, under the strongly monotonicity assumption, the regularized gap functions have fractional exponent error bounds, and thereby we provide an algorithm of Armijo type to solve the (VIP). / In this thesis, we investigate a nonsmooth variational inequality problem (VIP) defined by a locally Lipschitz function F which is not necessarily differentiable or monotone on its domain which is a closed convex set in an Euclidean space. / Tan Lulin. / "December 2005." / Adviser: Kung Fu Ng. / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6444. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 79-84) and index. / 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. / Abstracts in English and Chinese. / School code: 1307.

In this thesis,we report recent results on existence for variational inequalities in infinite-dimensional spaces under generalized monotonicity.

Electrical and electronic equipment waste (e-waste) is growing fast. Due to its potential economic value as well as its possible negative impacts on the environment, tracing e-waste flow is a major concern for stakeholders of e-waste management. Especially, whether or not adequate amounts of electrical and electronic equipment waste (WEEE) flow into the designed recycling systems is a fundamental issue for sustainable operations. In this paper, we analyze how technical, market, and legislative factors influence the total amount of e-waste that is collected, recycled, exported and (legally and illegally) disposed off. We formulate the e-waste network flow model as a variational inequality problem. The results of the numerical examples highlight the importance of considering the interaction between the supply and the demand side for precious materials in policy-decisions. Low collection rates of e-waste lead to low profits for stakeholders and make it difficult to establish sustainable recycling operations. Increasing WEEE collection rates increases recyclers' profits; however, it only increases smelters' profits up to a certain limit, after which smelters cannot benefit further due to limited demand for precious materials. Furthermore, the results emphasize the importance of establishing international control regimes for WEEE flows and reveal possible negative consequences of the recent trend of dematerialization. More precisely, product dematerialization tends to decrease recyles' and smelters' profits as well as to increase the outflow of e-waste from the designated recycling system. (authors' abstract)

This work aims to investigate some applications of the conjugate duality for scalar and vector optimization problems to the construction of gap functions for variational inequalities and equilibrium problems. The basic idea of the approach is to reformulate variational inequalities and equilibrium problems into optimization problems depending on a fixed variable, which allows us to apply duality results from optimization problems. Based on some perturbations, first we consider the conjugate duality for scalar optimization. As applications, duality investigations for the convex partially separable optimization problem are discussed. Afterwards, we concentrate our attention on some applications of conjugate duality for convex optimization problems in finite and infinite-dimensional spaces to the construction of a gap function for variational inequalities and equilibrium problems. To verify the properties in the definition of a gap function weak and strong duality are used. The remainder of this thesis deals with the extension of this approach to vector variational inequalities and vector equilibrium problems. By using the perturbation functions in analogy to the scalar case, different dual problems for vector optimization and duality assertions for these problems are derived. This study allows us to propose some set-valued gap functions for the vector variational inequality. Finally, by applying the Fenchel duality on the basis of weak orderings, some variational principles for vector equilibrium problems are investigated.

by Yan Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 78-80). / Abstract also in Chinese. / Chapter 1 --- Admissible Points of Convex Sets --- p.7 / Chapter 1.1 --- Introduction and Notations --- p.7 / Chapter 1.2 --- The Main Result --- p.7 / Chapter 1.2.1 --- The Proof of Theoreml.2.1 --- p.8 / Chapter 1.3 --- An Application --- p.10 / Chapter 2 --- A Generalization on The Theorems of Admissible Points --- p.12 / Chapter 2.1 --- Introduction and Notations --- p.12 / Chapter 2.2 --- Fundamental Lemmas --- p.14 / Chapter 2.3 --- The Main Result --- p.16 / Chapter 3 --- Introduction to Variational Inequalities --- p.21 / Chapter 3.1 --- Variational Inequalities in Finite Dimensional Space --- p.21 / Chapter 3.2 --- Problems Which Relate to Variational Inequalities --- p.25 / Chapter 3.3 --- Some Variations on Variational Inequality --- p.28 / Chapter 3.4 --- The Vector Variational Inequality Problem and Its Relation with The Vector Optimization Problem --- p.29 / Chapter 3.5 --- Variational Inequalities in Hilbert Space --- p.31 / Chapter 4 --- Vector Variational Inequalities --- p.36 / Chapter 4.1 --- Preliminaries --- p.36 / Chapter 4.2 --- Notations --- p.37 / Chapter 4.3 --- Existence Results of Vector Variational Inequality --- p.38 / Chapter 5 --- The Generalized Quasi-Variational Inequalities --- p.44 / Chapter 5.1 --- Introduction --- p.44 / Chapter 5.2 --- Properties of The Class F0 --- p.46 / Chapter 5.3 --- Main Theorem --- p.53 / Chapter 5.4 --- Remarks --- p.58 / Chapter 6 --- A set-valued open mapping theorem and related re- sults --- p.61 / Chapter 6.1 --- Introduction and Notations --- p.61 / Chapter 6.2 --- An Open Mapping Theorem --- p.62 / Chapter 6.3 --- Main Result --- p.63 / Chapter 6.4 --- An Application on Ordered Normed Spaces --- p.66 / Chapter 6.5 --- An Application on Open Decomposition --- p.70 / Chapter 6.6 --- An Application on Continuous Mappings from Order- infrabarreled Spaces --- p.72 / Bibliography