1 
Variational inequalities with the analytic center cutting plane methodDenault, M. (Michel) January 1998 (has links)
This thesis concerns the solution of variational inequalities (VIs) with analytic center cutting plane methods (ACCPMs). A convex feasibility problem reformulation of the variational inequality is used; this reformulation applies to VIs defined with pseudomonotone, singlevalued mappings or with maximal monotone, multivalued mappings. / Two cutting plane methods are presented: the first is based on linear cuts while the second uses quadratic cuts. The first method, ACCPMVI (linear cuts), requires mapping evaluations but no Jacobian evaluations; in fact, no differentiability assumption is needed. The cuts are placed at approximate analytic centers that are tracked with infeasible primaldual Newton steps. Linear equality constraints may be present in the definition of the VI's set of reference, and are treated explicitly. The set of reference is assumed to be polyhedral, or is convex and iteratively approximated by polyhedra. Alongside of the sequence of analytic centers, another sequence of points is generated, based on convex combinations of the analytic centers. This latter sequence is observed to converge to a solution much faster than the former sequence. / The second method, ACCPMVI (quadratic cuts), has cuts based on both mapping evaluations and Jacobian evaluations. The use of such a richer information set allows cuts that guide more accurately the sequence of analytic centers towards a solution. Mappings are assumed to be strongly monotone. However, Jacobian approximations, relying only on mapping evaluations, are observed to work very well in practice, so that differentiability of the mappings may not be required. There are two versions of the ACCPMVI (quadratic cuts), that differ in the way a new analytic center is reached after the introduction of a cut. One version uses a curvilinear search followed by dual Newton centering steps. The search entails a full eigenvectoreigenvalue decomposition of a dense matrix of the order of the number of variables. The other version uses two line searches, primaldual Newton steps, but no eigenvectoreigenvalue decomposition. / The algorithms described in this thesis were implemented in the M ATLAB environment. Numerical tests were performed on a variety of problems, some new and some traditional applications of variational inequalities.

2 
On the stability and propagation of barotropic modons in slowly varying mediaSwaters, Gordon Edwin January 1985 (has links)
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 eastwardtravelling modons. It is shown that eastwardtravelling and westwardtravelling 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 eastwardtravelling 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. Eastwardtravelling atmospheric modons are neutrally stable to the observed seasonally and annuallyaveraged atmospheric eddies. The neutral stability of westwardtravelling 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 eastwardtravelling 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 finiteamplitude 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 smallamplitude topography. The perturbations take the form of westward and eastwardtravelling transients and a stationary component proportional to the topography. The general solution is applied to ridgelike and escarpmentlike topographic configurations. / Science, Faculty of / Mathematics, Department of / Graduate

3 
Variational inequalities with the analytic center cutting plane methodDenault, M. (Michel) January 1998 (has links)
No description available.

4 
On merit functions and error bounds for variational inequality problem.January 2004 (has links)
Li GuoYin. / Thesis (M.Phil.)Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 105107). / 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  Dgap function  p.44 / Chapter 3.3  Generalized Regularize gap function and Generalized Dgap 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 Dgap function  p.78 / Chapter 4.3  Error bound results for Generalized Regularized gap function  p.92 / Chapter 4.4  Error bound results for Generalized Dgap function  p.93 / Bibliography  p.105

5 
Solving variational inequalities and related problems using recurrent neural networks. / CUHK electronic theses & dissertations collectionJanuary 2007 (has links)
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 KarushKuhnTucker (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: 6902, Section: B, page: 1102. / Thesis (Ph.D.)Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 193207). / 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.

6 
Nonmonotone Multivalued MappingsWang, Rongyi 02 June 2006 (has links)
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 singlevalued and multivalued mappings, we introduce the weak (S)_+ condition for singlevalued 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.

7 
Penalidades exatas para desigualdades variacionais / Exact Penalties for Variational InequalitiesThiago Afonso de Andre 01 February 2007 (has links)
Esta dissertação busca aproveitar os métodos de penalidades exatas diferenciáveis de programação nãolinear 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 nãolinear. 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 nãoexatas diferenciáveis , exatas nãodiferenciáveis 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.

8 
Penalidades exatas para desigualdades variacionais / Exact Penalties for Variational InequalitiesAndre, Thiago Afonso de 01 February 2007 (has links)
Esta dissertação busca aproveitar os métodos de penalidades exatas diferenciáveis de programação nãolinear 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 nãolinear. 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 nãoexatas diferenciáveis , exatas nãodiferenciáveis 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.

9 
On merit functions, error bounds, minimizing and stationary sequences for nonsmooth variational inequality problems. / CUHK electronic theses & dissertations collectionJanuary 2005 (has links)
First, we study the associated regularized gap functions and the Dgap functions and compute their ClarkeRockafellar 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 Dgap 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: 6711, Section: B, page: 6444. / Thesis (Ph.D.)Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 7984) 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.

10 
Variational InequalitiesHung, Shinyi 18 July 2007 (has links)
In this thesis,we report recent results on existence for variational inequalities in infinitedimensional spaces under generalized monotonicity.

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