Дисертації: "Generalized Nash equilibrium problems"

1

Sudermann-Merx, Nathan Georg [Verfasser], and O. [Akademischer Betreuer] Stein. "Linear Generalized Nash Equilibrium Problems / Nathan Georg Sudermann-Merx. Betreuer: O. Stein." Karlsruhe : KIT-Bibliothek, 2016. http://d-nb.info/1102250236/34.

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2

Heusinger, Anna von. "Numerical Methods for the Solution of the Generalized Nash Equilibrium Problem." kostenfrei, 2009. http://www.opus-bayern.de/uni-wuerzburg/volltexte/2010/4766/.

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3

Dreves, Axel [Verfasser], and Christian [Akademischer Betreuer] Kanzow. "Globally Convergent Algorithms for the Solution of Generalized Nash Equilibrium Problems / Axel Dreves. Betreuer: Christian Kanzow." Würzburg : Universitätsbibliothek der Universität Würzburg, 2012. http://d-nb.info/1020570881/34.

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4

Harms, Nadja [Verfasser], and Christian [Gutachter] Kanzow. "Primal and Dual Gap Functions for Generalized Nash Equilibrium Problems and Quasi-Variational Inequalities / Nadja Harms. Gutachter: Christian Kanzow." Würzburg : Universität Würzburg, 2014. http://d-nb.info/1102828769/34.

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5

Börgens, Eike Alexander Lars Guido [Verfasser], Christian [Gutachter] Kanzow, and Radu Ioan [Gutachter] Boţ. "ADMM-Type Methods for Optimization and Generalized Nash Equilibrium Problems in Hilbert Spaces / Eike Alexander Lars Guido Börgens ; Gutachter: Christian Kanzow, Radu Ioan Boţ." Würzburg : Universität Würzburg, 2020. http://d-nb.info/1223851370/34.

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6

Rojas, Frank Navarro. "Condições de otimalidade, qualificação e métodos tipo Lagrangiano aumentado para problemas de equilíbrio de Nash generalizados." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-27032018-114413/.

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Esta tese é um estudo acerca do Problema de Equilíbrio de Nash Generalizado (GNEP). Na primeira parte, faremos um resumo dos principais conceitos sobre GNEPs, a relação com outros problemas já conhecidos e comentaremos brevemente os principais métodos já feitos até esta data para resolver numericamente este tipo de problema. Na segunda parte, estudamos condições de otimalidade e condições de qualificação (CQ) para GNEPs, fazendo uma analogia como em otimização. Estendemos os conceitos de cone tangente, normal, gerado pelas restrições ativas, linearizado e polar para a estrutura dos GNEPs. Cada CQ de otimização gera dois tipos de CQ para GNEPs, sendo que a denotada por CQ-GNEP é mais forte e útil para a análise de algoritmos para GNEPs. Mostramos que as condições de qualificação para GNEPs deste tipo em alguns casos não guardam a mesma relação que em otimização. Estendemos também o conceito de Aproximadamente Karush-KuhnTucker (AKKT) de otimização para GNEPs, o AKKT-GNEP. É bem conhecido que AKKT é uma genuína condição de otimalidade em otimização, mas para o caso dos GNEPs mostramos que isto não ocorre em geral. Por outro lado, AKKT-GNEP é satisfeito, por exemplo, em qualquer solução de um GNEP conjuntamente convexo, desde que seja um equilíbrio bvariacional. Com isso em mente, definimos um método do tipo Lagrangiano Aumentado para o GNEP usando penalidades quadráticas e exponenciais e estudamos as propriedades de otimalidade e viabilidade dos pontos limites de sequências geradas pelo algoritmo. Finalmente alguns critérios para resolver os subproblemas e resultados numéricos são apresentados. This thesis is a study about the generalized Nash equilibrium problem (GNEP). In the first part we will summarize the main concepts about GNEPs, the relationship with other known problems and we will briefly comment on the main methods already done in order to solve these problems numerically. In the second part we study optimality conditions and constraint qualification (CQ) for GNEPs making an analogy with the optimization case. We extend the concepts of the tangent, normal and generated by the active cones, linear and polar cone to the structure of the GNEPs. Each optimization CQ generates two types of CQs for GNEPs, with the one called CQ-GNEP being the strongest and most useful for analyzing the algorithms for GNEPs. We show that the qualification conditions for GNEPs of this type in some cases do not have the same relation as in optimization. We also extend the Approximate Karush- Kuhn-Tucker (AKKT) concept used in optimization for GNEPs to AKKT-GNEP. It is well known that AKKT is a genuine optimality condition in optimization but for GNEPs we show that this does not occur in general. On the other hand, AKKT-GNEP is satisfied, for example, in any solution of a jointly convex GNEP, provided that it is a b-variational equilibrium. With this in mind, we define Augmented Lagrangian methods for the GNEP, using the quadratic and the exponential penalties, and we study the optimality and feasibility properties of the sequence of points generated by the algorithms. Finally some criteria to solve the subproblems and numerical results are presented.

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7

Herrich, Markus. "Local Convergence of Newton-type Methods for Nonsmooth Constrained Equations and Applications." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-159569.

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In this thesis we consider constrained systems of equations. The focus is on local Newton-type methods for the solution of constrained systems which converge locally quadratically under mild assumptions implying neither local uniqueness of solutions nor differentiability of the equation function at solutions. The first aim of this thesis is to improve existing local convergence results of the constrained Levenberg-Marquardt method. To this end, we describe a general Newton-type algorithm. Then we prove local quadratic convergence of this general algorithm under the same four assumptions which were recently used for the local convergence analysis of the LP-Newton method. Afterwards, we show that, besides the LP-Newton method, the constrained Levenberg-Marquardt method can be regarded as a special realization of the general Newton-type algorithm and therefore enjoys the same local convergence properties. Thus, local quadratic convergence of a nonsmooth constrained Levenberg-Marquardt method is proved without requiring conditions implying the local uniqueness of solutions. As already mentioned, we use four assumptions for the local convergence analysis of the general Newton-type algorithm. The second aim of this thesis is a detailed discussion of these convergence assumptions for the case that the equation function of the constrained system is piecewise continuously differentiable. Some of the convergence assumptions seem quite technical and difficult to check. Therefore, we look for sufficient conditions which are still mild but which seem to be more familiar. We will particularly prove that the whole set of the convergence assumptions holds if some set of local error bound conditions is satisfied and in addition the feasible set of the constrained system excludes those zeros of the selection functions which are not zeros of the equation function itself, at least in a sufficiently small neighborhood of some fixed solution. We apply our results to constrained systems arising from complementarity systems, i.e., systems of equations and inequalities which contain complementarity constraints. Our new conditions are discussed for a suitable reformulation of the complementarity system as constrained system of equations by means of the minimum function. In particular, it will turn out that the whole set of the convergence assumptions is actually implied by some set of local error bound conditions. In addition, we provide a new constant rank condition implying the whole set of the convergence assumptions. Particularly, we provide adapted formulations of our new conditions for special classes of complementarity systems. We consider Karush-Kuhn-Tucker (KKT) systems arising from optimization problems, variational inequalities, or generalized Nash equilibrium problems (GNEPs) and Fritz-John (FJ) systems arising from GNEPs. Thus, we obtain for each problem class conditions which guarantee local quadratic convergence of the general Newton-type algorithm and its special realizations to a solution of the particular problem. Moreover, we prove for FJ systems of GNEPs that generically some full row rank condition is satisfied at any solution of the FJ system of a GNEP. The latter condition implies the whole set of the convergence assumptions if the functions which characterize the GNEP are sufficiently smooth. Finally, we describe an idea for a possible globalization of our Newton-type methods, at least for the case that the constrained system arises from a certain smooth reformulation of the KKT system of a GNEP. More precisely, a hybrid method is presented whose local part is the LP-Newton method. The hybrid method turns out to be, under appropriate conditions, both globally and locally quadratically convergent.

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8

Batista, Edvaldo Elias de Almeida. "Generalized vector equilibrium problems and algorithms for variational inequality in hadamard manifolds." Universidade Federal de Goiás, 2016. http://repositorio.bc.ufg.br/tede/handle/tede/6562.

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Submitted by Jaqueline Silva (jtas29@gmail.com) on 2016-12-09T17:10:49Z No. of bitstreams: 2 Tese - Edvaldo Elias de Almeida Batista - 2016.pdf: 1198471 bytes, checksum: 88d7db305f0cfe6be9b62496a226217f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-12-09T17:11:03Z (GMT) No. of bitstreams: 2 Tese - Edvaldo Elias de Almeida Batista - 2016.pdf: 1198471 bytes, checksum: 88d7db305f0cfe6be9b62496a226217f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Made available in DSpace on 2016-12-09T17:11:03Z (GMT). No. of bitstreams: 2 Tese - Edvaldo Elias de Almeida Batista - 2016.pdf: 1198471 bytes, checksum: 88d7db305f0cfe6be9b62496a226217f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-10-20 Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES In this thesis, we study variational inequalities and generalized vector equilibrium problems. In Chapter 1, several results and basic deﬁnitions of Riemannian geometry are listed; we present the concept of the monotone vector ﬁeld in Hadamard manifolds and many of their properties, besides, we introduce the concept of enlargement of a monotone vector ﬁeld, and we display its properties in a Riemannian context. In Chapter 2, an inexact proximal point method for variational inequalities in Hadamard manifolds is introduced, and its convergence properties are studied; see [7]. To present our method, we generalize the concept of enlargement of monotone operators, from a linear setting to the Riemannian context. As an application, an inexact proximal point method for constrained optimization problems is obtained. In Chapter 3, we present an extragradient algorithm for variational inequality associated with the point-to-set vector ﬁeld in Hadamard manifolds and study its convergence properties; see [8]. In order to present our method, the concept of enlargement of maximal monotone vector ﬁelds is used and its lower-semicontinuity is established to obtain the convergence of the method in this new context. In Chapter 4, we present a suﬃcient condition for the existence of a solution to the generalized vector equilibrium problem on Hadamard manifolds using a version of the KnasterKuratowski-Mazurkiewicz Lemma; see [6]. In particular, the existence of solutions to optimization, vector optimization, Nash equilibria, complementarity, and variational inequality is a special case of the existence result for the generalized vector equilibrium problem. Nesta tese, estudamos desigualdades variacionais e o problema de equilíbrio vetorial generalizado. No Capítulo 1, vários resultados e definições elementares sobre geometria Riemanniana são enunciados; apresentamos o conceito de campo vetorial monótono e muitas de suas propriedades, além de introduzir o conceito de alargamento de um campo vetorial monótono e exibir suas propriedades em um contexto Riemanniano. No Capítulo 2, um método de ponto proximal inexato para desigualdades variacionais em variedades de Hadamard é introduzido e suas propriedades de convergência são estudadas; veja [7]. Para apresentar o nosso método, generalizamos o conceito de alargamento de operadores monótonos, do contexto linear ao contexto de Riemanniano. Como aplicação, é obtido um método de ponto proximal inexato para problemas de otimização com restrições. No Capítulo 3, apresentamos um algoritmo extragradiente para desigualdades variacionais associado a um campo vetorial ponto-conjunto em variedades de Hadamard e estudamos suas propriedades de convergência; veja [8]. A fim de apresentar nosso método, o conceito de alargamento de campos vetoriais monótonos é utilizado e sua semicontinuidade inferior é estabelecida, a fim de obter a convergência do método neste novo contexto. No Capítulo 4, apresentamos uma condição suficiente para a existência de soluções para o problema de equilíbrio vetorial generalizado em variedades de Hadamard usando uma versão do Lema Knaster-Kuratowski-Mazurkiewicz; veja [6]. Em particular, a existência de soluções para problemas de otimização, otimização vetorial, equilíbrio de Nash, complementaridade e desigualdades variacionais são casos especiais do resultado de existência do problema de equilíbrio vetorial generalizado.

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9

Ye, Zhineng. "Solving Eight Treasures Of Game Theory Problems Using Bi-criteria Method." Case Western Reserve University School of Graduate Studies / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=case1454062652.

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10

Abada, Ibrahim. "Modélisation des marchés du gaz naturel en Europe en concurrence oligopolistique : le modèle GaMMES et quelques applications." Thesis, Paris 10, 2012. http://www.theses.fr/2012PA100043/document.

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Анотація:

Cette thèse étudie l’évolution des marchés du gaz naturel en Europe jusqu’en 2035 en utilisant les outils de la modélisation. Le modèle proposé, intitulé GaMMES, repose sur une description oligopolistique des marchés et ses principaux avantages sont les suivants : un niveau de détail important de la structure économique de la chaîne gazière et une prise en compte endogène des contrats de long-terme en amont ainsi que de la substitution avec les produits pétroliers et le charbon, au niveau de la demande. Dans un premier temps, nous étudions la question de la sécurité d’approvisionnement en gaz en Europe et les conditions favorables à la régulation des marchés vulnérables au risque de rupture d’approvisionnement, notamment de la part de la Russie. Trois études de cas sont proposées selon le degré de dépendance et la nature de régulation en place : le marché allemand des années 1980 et les marchés actuels de la Bulgarie et de l’Espagne. Nous étudions en particulier l’évolution des caractéristiques des marchés en fonction du risque de rupture et le type de régulation à mettre en place afin d’assurer l’optimalité du bien-être social. Ensuite, nous proposons un modèle de type systèmes dynamiques afin de prendre en compte la substitution énergétique entre le charbon, le pétrole et le gaz naturel. Notre approche permet d’estimer une nouvelle forme fonctionnelle de la fonction de demande pour le gaz naturel, qui englobe à la fois la substitution énergétique et les inerties de consommation dues aux investissements des usagers finaux. Dans un troisième temps, nous utilisons cette fonction de demande dans un modèle d’équilibre partiel des marchés du gaz naturel en Europe. Le modèle GaMMES, écrit sous forme de problème de complémentarité, représente les principaux acteurs de l’industrie du gaz naturel en considérant leurs interactions stratégiques et les pouvoirs de marchés. Il a été appliqué au marché du gaz naturel en Europe du nord-est afin d’étudier l’évolution, jusqu’en 2035, de la consommation, des prix spot, des prix et volumes long-terme, de la production et de la dépendance par rapport aux imports étrangers. Finalement, nous proposons une extension stochastique du modèle GaMMES afin d’analyser l’impact de la forte fluctuation du prix du Brent sur les marchés gaziers. Une étude économétrique a été menée afin de calculer la loi de probabilité du prix du pétrole, lorsqu’il est modélisé en tant que variable aléatoire, dans le but de construire et pondérer l’arbre des scénarii. Les résultats permettent de comprendre comment l’aléa modifie les comportements stratégiques des acteurs, notamment au niveau des contrats de long-terme. Enfin, la valeur de la solution stochastique est calculée afin de quantifier l’importance de la prise en compte des fluctuations du prix du pétrole pour chaque acteur de la chaîne This thesis studies the evolution of the natural gas markets in Europe, until 2035, using optimization theory tools. The model we develop, named GaMMES, is based on an oligopolistic description of the markets. Its main advantages are the following: we consider an important level of detail in the economic structure of the gas chain and we endogenously take into account long-term contracts in the upstream as well as energy substitution between gas, oil, and coal in the demand. In the first part of this thesis, we study the issue of security of supply in Europe and the conditions under which it is necessary to regulate the gas markets that are strongly dependent on foreign imports. Three case studies are then presented, regarding the level of dependence and the markets' specificities: the German gas trade of the 1980s and the current Spanish and Bulgarian markets. We study in particular the evolution of the markets' outcome as a function of the supply disruption probability and the kind of regulation to implement in order to maximize the social welfare. In the second part, we develop a system dynamics model in order to capture fuel substitution between oil, coal, and natural gas. Our approach allows one to calculate a new functional form of the demand function for natural gas that contains energy substitution and consumption inertia effects due to end-users' investments. In the third part, we take advantage of our demand function and use it in a partial equilibrium model of natural gas markets in Europe. The GaMMES model, when written as a complementarity problem, describes the principal gas chain actors as well as their strategic interactions and market power. It was applied to the northwestern European gas trade to analyze the evolution of consumption, spot and long-term contract prices and volumes, production, and natural gas dependence, until 2035. In the last part, we present a stochastic extension of the GaMMES model in order to study the impact of the strong Brent price fluctuation on the gas markets. An econometric analysis allowed us to calculate the probability law of the oil price, when taken as a random variable, in order to construct the scenario tree and estimate its weights. Our results show how uncertainty changes the strategic behavior, in particular for the long-term contracting activity. Finally, the value of the stochastic solution is calculated to quantify the importance of taking into account randomness in the optimization programs of the gas chain actors

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11

Svensson, Anton. "Non-smooth and variational analysis of optimization problems and multi-leader-follower games." Thesis, Perpignan, 2020. http://www.theses.fr/2020PERP0003.

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Cette thèse, dont le cadre général est l'optimisation, traite de problèmes d'optimisation non-lisse et de problèmes de théorie des jeux. Elle est constituée de quatre parties. Dans la première, nous présentons le contexte et l'introduction. Dans la deuxième partie, nous discutons quelques règles de calcul sous-différentiel dans des espaces généraux, et présentons notamment certaines formules plus fortes que l'état de l'art, autant dans le cas convexe que dans le cas non convexe. L'accent est mis sur les règles de calcul et conditions d'optimalité approchées et "fuzzy", pour lesquelles aucune condition de qualification n'est requise. Dans la troisième partie, nous considérons des jeux bi-niveaux à plusieurs meneurs et plusieurs suiveurs. Après quelques résultats d'existence dans le cas d'un seul meneur optimiste et dans le cas de plusieurs meneurs, nous étendons des résultats existants concernant la relation entre le problème bi-niveau original et sa reformulation obtenue grâce au remplacement des problèmes des suiveurs par la concaténation de leurs conditions d'optimalité (KKT). Finalement, dans la quatrième partie, nous abordons quelques problèmes de quasi-équilibre, qui sont une généralisation des problèmes d'équilibre de Nash et des inégalités quasi-variationnelles. Nous prouvons ainsi de nouveaux résultats d'existence qui permettent de relâcher les hypothèses standard This thesis is within the framework of optimization and deals with nonsmooth optimization and with some problems of game theory. It is divided into four parts. In the first introductory part, we give the context and some preliminary results. In the second part we discuss about subdifferential calculus rules in general spaces providing of some improved formulas in both the convex and the non-convex cases. Here the focus is on approximate or fuzzy calculus rules and optimality conditions, for which no qualification conditions are required. In the third part, we discuss about the so-called Multi-Leader-Follower Games. We give an existence result for the case of a single optimistic leader and multiple followers, and extend some results concerning the relation between the original problem with the reformulation obtained by replacing the followers' problem by the concatenation of their KKT conditions. Finally, in the fourth part we study quasi-equilibrium problems which are a general formulation for studying Nash equilibrium problems and quasi-variational inequalities. We provide some new existence results that relax some of the standard hypotheses

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12

Dutang, Christophe. "Etude des marchés d'assurance non-vie à l'aide d'équilibre de Nash et de modèle de risques avec dépendance." Phd thesis, Université Claude Bernard - Lyon I, 2012. http://tel.archives-ouvertes.fr/tel-00703797.

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L'actuariat non-vie étudie les différents aspects quantitatifs de l'activité d'assurance. Cette thèse vise à expliquer sous différentes perspectives les interactions entre les différents agents économiques, l'assuré, l'assureur et le marché, sur un marché d'assurance. Le chapitre 1 souligne à quel point la prise en compte de la prime marché est importante dans la décision de l'assuré de renouveler ou non son contrat d'assurance avec son assureur actuel. La nécessitéd'un modèle de marché est établie. Le chapitre 2 répond à cette problématique en utilisant la théorie des jeux non-coopératifs pour modéliser la compétition. Dans la littérature actuelle, les modèles de compétition seréduisent toujours à une optimisation simpliste du volume de prime basée sur une vision d'un assureur contre le marché. Partant d'un modèle de marché à une période, un jeu d'assureurs est formulé, où l'existence et l'unicité de l'équilibre de Nash sont vérifiées. Les propriétés des primes d'équilibre sont étudiées pour mieux comprendre les facteurs clés d'une position dominante d'un assureur par rapport aux autres. Ensuite, l'intégration du jeu sur une période dans un cadre dynamique se fait par la répétition du jeu sur plusieurs périodes. Une approche par Monte-Carlo est utilisée pour évaluer la probabilité pour un assureur d'être ruiné, de rester leader, de disparaître du jeu par manque d'assurés en portefeuille. Ce chapitre vise à mieux comprendre la présence de cycles en assurance non-vie. Le chapitre 3 présente en profondeur le calcul effectif d'équilibre de Nash pour n joueurs sous contraintes, appelé équilibre de Nash généralisé. Il propose un panorama des méthodes d'optimisation pour la résolution des n sous-problèmes d'optimisation. Cette résolution sefait à l'aide d'une équation semi-lisse basée sur la reformulation de Karush-Kuhn-Tucker duproblème d'équilibre de Nash généralisé. Ces équations nécessitent l'utilisation du Jacobiengénéralisé pour les fonctions localement lipschitziennes intervenant dans le problème d'optimisation.Une étude de convergence et une comparaison des méthodes d'optimisation sont réalisées.Enfin, le chapitre 4 aborde le calcul de la probabilité de ruine, un autre thème fondamentalde l'assurance non-vie. Dans ce chapitre, un modèle de risque avec dépendance entre lesmontants ou les temps d'attente de sinistre est étudié. De nouvelles formules asymptotiquesde la probabilité de ruine en temps infini sont obtenues dans un cadre large de modèle de risquesavec dépendance entre sinistres. De plus, on obtient des formules explicites de la probabilité deruine en temps discret. Dans ce modèle discret, l'analyse structure de dépendance permet dequantifier l'écart maximal sur les fonctions de répartition jointe des montants entre la versioncontinue et la version discrète.

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13

Theljani, Anis. "Partial differential equations methods and regularization techniques for image inpainting." Thesis, Mulhouse, 2015. http://www.theses.fr/2015MULH0278/document.

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Cette thèse concerne le problème de désocclusion d'images, au moyen des équations aux dérivées partielles. Dans la première partie de la thèse, la désocclusion est modélisée par un problème de Cauchy qui consiste à déterminer une solution d'une équation aux dérivées partielles avec des données aux bords accessibles seulement sur une partie du bord de la partie à recouvrir. Ensuite, on a utilisé des algorithmes de minimisation issus de la théorie des jeux, pour résoudre ce problème de Cauchy. La deuxième partie de la thèse est consacrée au choix des paramètres de régularisation pour des EDP d'ordre deux et d'ordre quatre. L'approche développée consiste à construire une famille de problèmes d'optimisation bien posés où les paramètres sont choisis comme étant une fonction variable en espace. Ceci permet de prendre en compte les différents détails, à différents échelles dans l'image. L'apport de la méthode est de résoudre de façon satisfaisante et objective, le choix du paramètre de régularisation en se basant sur des indicateurs d'erreur et donc le caractère à posteriori de la méthode (i.e. indépendant de la solution exacte, en générale inconnue). En outre, elle fait appel à des techniques classiques d'adaptation de maillage, qui rendent peu coûteuses les calculs numériques. En plus, un des aspects attractif de cette méthode, en traitement d'images est la récupération et la détection de contours et de structures fines Image inpainting refers to the process of restoring a damaged image with missing information. Different mathematical approaches were suggested to deal with this problem. In particular, partial differential diffusion equations are extensively used. The underlying idea of PDE-based approaches is to fill-in damaged regions with available information from their surroundings. The first purpose of this Thesis is to treat the case where this information is not available in a part of the boundary of the damaged region. We formulate the inpainting problem as a nonlinear boundary inverse problem for incomplete images. Then, we give a Nash-game formulation of this Cauchy problem and we present different numerical which show the efficiency of the proposed approach as an inpainting method.Typically, inpainting is an ill-posed inverse problem for it most of PDEs approaches are obtained from minimization of regularized energies, in the context of Tikhonov regularization. The second part of the thesis is devoted to the choice of regularization parameters in second-and fourth-order energy-based models with the aim of obtaining as far as possible fine features of the initial image, e.g., (corners, edges, … ) in the inpainted region. We introduce a family of regularized functionals with regularization parameters to be selected locally, adaptively and in a posteriori way allowing to change locally the initial model. We also draw connections between the proposed method and the Mumford-Shah functional. An important feature of the proposed method is that the investigated PDEs are easy to discretize and the overall adaptive approach is easy to implement numerically

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14

Dreves, Axel. "Globally Convergent Algorithms for the Solution of Generalized Nash Equilibrium Problems." Doctoral thesis, 2011. https://nbn-resolving.org/urn:nbn:de:bvb:20-opus-69822.

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Es werden verschiedene Verfahren zur Lösung verallgemeinerter Nash-Gleichgewichtsprobleme mit dem Schwerpunkt auf deren globaler Konvergenz entwickelt. Ein globalisiertes Newton-Verfahren zur Berechnung normalisierter Lösungen, ein nichtglattes Optimierungsverfahren basierend auf einer unrestringierten Umformulierung des spieltheoretischen Problems, und ein Minimierungsansatz sowei eine Innere-Punkte-Methode zur Lösung der gemeinsamen Karush-Kuhn-Tucker-Bedingungen der Spieler werden theoretisch untersucht und numerisch getestet. Insbesondere das Innere-Punkte Verfahren erweist sich als das zur Zeit wohl beste Verfahren zur Lösung verallgemeinerter Nash-Gleichgewichtsprobleme In this thesis different algorithms for the solution of generalized Nash equilibrium problems with the focus on global convergence properties are developed. A globalized Newton method for the computation of normalized solutions, a nonsmooth algorithm based on an optimization reformulation of the game-theoretic problem, and a merit function approach and an interior point method for the solution of the concatenated Karush-Kuhn-Tucker-system are analyzed theoretically and numerically. The interior point method turns out to be one of the best existing methods for the solution of generalized Nash equilibrium problems

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15

Börgens, Eike Alexander Lars Guido. "ADMM-Type Methods for Optimization and Generalized Nash Equilibrium Problems in Hilbert Spaces." Doctoral thesis, 2020. https://doi.org/10.25972/OPUS-21877.

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This thesis is concerned with a certain class of algorithms for the solution of constrained optimization problems and generalized Nash equilibrium problems in Hilbert spaces. This class of algorithms is inspired by the alternating direction method of multipliers (ADMM) and eliminates the constraints using an augmented Lagrangian approach. The alternating direction method consists of splitting the augmented Lagrangian subproblem into smaller and more easily manageable parts. Before the algorithms are discussed, a substantial amount of background material, including the theory of Banach and Hilbert spaces, fixed-point iterations as well as convex and monotone set-valued analysis, is presented. Thereafter, certain optimization problems and generalized Nash equilibrium problems are reformulated and analyzed using variational inequalities and set-valued mappings. The analysis of the algorithms developed in the course of this thesis is rooted in these reformulations as variational inequalities and set-valued mappings. The first algorithms discussed and analyzed are one weakly and one strongly convergent ADMM-type algorithm for convex, linearly constrained optimization. By equipping the associated Hilbert space with the correct weighted scalar product, the analysis of these two methods is accomplished using the proximal point method and the Halpern method. The rest of the thesis is concerned with the development and analysis of ADMM-type algorithms for generalized Nash equilibrium problems that jointly share a linear equality constraint. The first class of these algorithms is completely parallelizable and uses a forward-backward idea for the analysis, whereas the second class of algorithms can be interpreted as a direct extension of the classical ADMM-method to generalized Nash equilibrium problems. At the end of this thesis, the numerical behavior of the discussed algorithms is demonstrated on a collection of examples Die vorliegende Arbeit behandelt eine Klasse von Algorithmen zur Lösung restringierter Optimierungsprobleme und verallgemeinerter Nash-Gleichgewichtsprobleme in Hilberträumen. Diese Klasse von Algorithmen ist angelehnt an die Alternating Direction Method of Multipliers (ADMM) und eliminiert die Nebenbedingungen durch einen Augmented-Lagrangian-Ansatz. Im Rahmen dessen wird in der Alternating Direction Method of Multipliers das jeweilige Augmented-Lagrangian-Teilproblem in kleinere Teilprobleme aufgespaltet. Zur Vorbereitung wird eine Vielzahl grundlegender Resultate präsentiert. Dies beinhaltet entsprechende Ergebnisse aus der Literatur zu der Theorie von Banach- und Hilberträumen, Fixpunktmethoden sowie konvexer und monotoner mengenwertiger Analysis. Im Anschluss werden gewisse Optimierungsprobleme sowie verallgemeinerte Nash-Gleichgewichtsprobleme als Variationsungleichungen und Inklusionen mit mengenwertigen Operatoren formuliert und analysiert. Die Analysis der im Rahmen dieser Arbeit entwickelten Algorithmen bezieht sich auf diese Reformulierungen als Variationsungleichungen und Inklusionsprobleme. Zuerst werden ein schwach und ein stark konvergenter paralleler ADMM-Algorithmus zur Lösung von separablen Optimierungsaufgaben mit linearen Gleichheitsnebenbedingungen präsentiert und analysiert. Durch die Ausstattung des zugehörigen Hilbertraums mit dem richtigen gewichteten Skalarprodukt gelingt die Analyse dieser beiden Methoden mit Hilfe der Proximalpunktmethode und der Halpern-Methode. Der Rest der Arbeit beschäftigt sich mit Algorithmen für verallgemeinerte Nash-Gleichgewichtsprobleme, die gemeinsame lineare Gleichheitsnebenbedingungen besitzen. Die erste Klasse von Algorithmen ist vollständig parallelisierbar und es wird ein Forward-Backward-Ansatz für die Analyse genutzt. Die zweite Klasse von Algorithmen kann hingegen als direkte Erweiterung des klassischen ADMM-Verfahrens auf verallgemeinerte Nash-Gleichgewichtsprobleme aufgefasst werden. Abschließend wird das Konvergenzverhalten der entwickelten Algorithmen an einer Sammlung von Beispielen demonstriert

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16

Harms, Nadja. "Primal and Dual Gap Functions for Generalized Nash Equilibrium Problems and Quasi-Variational Inequalities." Doctoral thesis, 2014. https://nbn-resolving.org/urn:nbn:de:bvb:20-opus-106027.

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In this thesis we study smoothness properties of primal and dual gap functions for generalized Nash equilibrium problems (GNEPs) and finite-dimensional quasi-variational inequalities (QVIs). These gap functions are optimal value functions of primal and dual reformulations of a corresponding GNEP or QVI as a constrained or unconstrained optimization problem. Depending on the problem type, the primal reformulation uses regularized Nikaido-Isoda or regularized gap function approaches. For player convex GNEPs and QVIs of the so-called generalized `moving set' type the respective primal gap functions are continuously differentiable. In general, however, these primal gap functions are nonsmooth for both problems. Hence, we investigate their continuity and differentiability properties under suitable assumptions. Here, our main result states that, apart from special cases, all locally minimal points of the primal reformulations are points of differentiability of the corresponding primal gap function. Furthermore, we develop dual gap functions for a class of GNEPs and QVIs and ensuing unconstrained optimization reformulations of these problems based on an idea by Dietrich (``A smooth dual gap function solution to a class of quasivariational inequalities'', Journal of Mathematical Analysis and Applications 235, 1999, pp. 380--393). For this purpose we rewrite the primal gap functions as a difference of two strongly convex functions and employ the Toland-Singer duality theory. The resulting dual gap functions are continuously differentiable and, under suitable assumptions, have piecewise smooth gradients. Our theoretical analysis is complemented by numerical experiments. The solution methods employed make use of the first-order information established by the aforementioned theoretical investigations In dieser Dissertation wurden die Glattheitseigenschaften von primalen und dualen Gap-Funktionen für verallgemeinerte Nash-Gleichgewichtsprobleme (GNEPs) und Quasi-Variationsungleichungen (QVIs) untersucht. Diese Gap-Funktionen sind Optimalwertfunktionen von primalen und dualen Umformulierungen eines GNEPs oder QVIs als restringiertes oder unrestringiertes Optimierungsproblem. Für gewisse Teilklassen von GNEPs (Spezialfall von `player convex' GNEPs) und QVIs (`generalized moving set case') sind diese primalen Gap-Funktionen überall stetig differenzierbar, für allgemeine GNEPs und QVIs jedoch nicht. Weitere Untersuchungen der Stetigkeit und Differenzierbarkeit ergaben, dass die primalen Gap-Funktionen unter geeigneten Bedingungen, abgesehen von Sonderfällen, in allen lokalen Minima der entsprechenden primalen Umformulierung differenzierbar sind. In dieser Dissertation wurden außerdem duale Gap-Funktionen für bestimmte Klassen von GNEPs und QVIs entwickelt, indem die primalen Gap-Funktionen basierend auf einer Idee von Dietrich (H. Dietrich: A smooth dual gap function solution to a class of quasivariational inequalities. Journal of Mathematical Analysis and Applications 235, 1999, pp. 380--393) als Differenz zweier gleichmäßig konvexer Funktionen dargestellt wurden und auf diese beiden Funktionen die Toland-Singer-Dualitätstheorie angewendet wurde. Es stellte sich heraus, dass diese dualen Gap-Funktionen stetig differenzierbar sind und unter geeigneten Bedingungen sogar stückweise stetig differenzierbare Gradienten besitzen. Die Ergebnisse in dieser Dissertation wurden durch numerische Berechnungen für diverse Testprobleme mittels bekannter Optimierungsverfahren erster Ordnung unterstützt

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von, Heusinger Anna. "Numerical Methods for the Solution of the Generalized Nash Equilibrium Problem." Doctoral thesis, 2009. https://nbn-resolving.org/urn:nbn:de:bvb:20-opus-47662.

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In the generalized Nash equilibrium problem not only the cost function of a player depends on the rival players' decisions, but also his constraints. This thesis presents different iterative methods for the numerical computation of a generalized Nash equilibrium, some of them globally, others locally superlinearly convergent. These methods are based on either reformulations of the generalized Nash equilibrium problem as an optimization problem, or on a fixed point formulation. The key tool for these reformulations is the Nikaido-Isoda function. Numerical results for various problem from the literature are given Das verallgemeinerte Nash-Gleichgewichtsproblem ist ein Lösungskonzept für Spiele, in denen neben der Kostenfunktion eines Spielers auch dessen Strategiemenge von den Entscheidungen der anderen Spieler abhängt. In dieser Arbeit werden global konvergente und lokal superlinear konvergente Verfahren zur numerischen Berechnung eines verallgemeinerten Nash-Gleichgewichts vorgestellt. Die Verfahren basieren entweder auf einer Umformulierung des verallgemeinerten Nash-Gleichgewichtsproblems als Optimierungsproblem oder als Fixpunktproblem. Für diese Umformulierungen wird die Nikaido-Isoda Funktion verwendet. Es werden numerische Ergebenisse für einige Probleme aus der Literatur widergegeben

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18

LAMPARIELLO, LORENZO. "Penalty methods for the solution of generalized Nash equilibrium problems and hemivariational inequalities with VI constraints." Doctoral thesis, 2012. http://hdl.handle.net/11573/918565.

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In this thesis we propose penalty methods for the solution of Generalized Nash Equilibrium Problems (GNEPs) and we consider centralized and distributed algorithms for the solution of Hemivariational Inequalities (HVIs) where the feasible set is given by the intersection of a closed convex set with the solution set of a lower-level monotone Variational Inequality (VI).

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19

Heusinger, Anna von [Verfasser]. "Numerical methods for the solution of the generalized Nash equilibrium problem / Anna von Heusinger." 2009. http://d-nb.info/1001800753/34.

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20

Galli, Leonardo. "Nonmonotone techniques for smooth optimization." Doctoral thesis, 2020. http://hdl.handle.net/2158/1202158.

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The focus of this thesis is the study and the application of nonmonotone strategies. These techniques are basically introduced to improve numerical results of existing optimization algorithms. Their first aim is that of relaxing the monotone requirement imposed by the globalization techniques. In fact, these monotone conditions might slow down the convergence rate of inher- ently nonmonotone optimization methods. This relaxation must not harm global convergence results. In this thesis we apply nonmonotone strategies to both line search and trust-region globalization techniques. We first considered Generalized Nash Equilibrium Problems (GNEPs) and their KKT reformulation into a highly nonlinear constrained smooth system of equations. In order to obtain global and fast local convergence, we take into account an existing trust-region method that is locally superlinear under an error bound condition only. A nonmonotone strategy has been applied, showing that the resulting algo- rithm performs significantly better than the original one. Global conver- gence properties have been proved for the new algorithm, while superlinear convergence is directly inherited from the existing method. The resulting algorithm is competitive with a standard software for nonlinear equations, not only on GNEPs, but also on quasi-variational inequalities. The second contribution of this thesis is the development of a framework for nonmonotone line search based decomposition methods. This is the first time in which nonmonotonicity is combined with decomposition methods for general constrained problems. Note that the choice of the direction and the line search are not fixed in advance, in fact the framework proves conver- gence for all those combinations of directions and line searches that are able to satisfy some mild assumptions. A specific realization of this abstract algo- rithm has been implemented in two versions, monotone and nonmonotone. The two algorithms have been compared on a set of network equilibrium problems. Also on this application, the nonmonotone version outperformed its monotone counterpart both on the total number of iterations and the function evaluations. In the end, a new family of nonmonotone techniques is proposed to build algorithms that are able to control the amount of nonmonotonicity intro- duced in each of the phases of the optimization procedure. This tool might be very helpful to understand in which combination of methods, problems and phases is more important to apply a nonmonotone strategy.

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21

Herrich, Markus. "Local Convergence of Newton-type Methods for Nonsmooth Constrained Equations and Applications." Doctoral thesis, 2014. https://tud.qucosa.de/id/qucosa%3A28495.

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In this thesis we consider constrained systems of equations. The focus is on local Newton-type methods for the solution of constrained systems which converge locally quadratically under mild assumptions implying neither local uniqueness of solutions nor differentiability of the equation function at solutions. The first aim of this thesis is to improve existing local convergence results of the constrained Levenberg-Marquardt method. To this end, we describe a general Newton-type algorithm. Then we prove local quadratic convergence of this general algorithm under the same four assumptions which were recently used for the local convergence analysis of the LP-Newton method. Afterwards, we show that, besides the LP-Newton method, the constrained Levenberg-Marquardt method can be regarded as a special realization of the general Newton-type algorithm and therefore enjoys the same local convergence properties. Thus, local quadratic convergence of a nonsmooth constrained Levenberg-Marquardt method is proved without requiring conditions implying the local uniqueness of solutions. As already mentioned, we use four assumptions for the local convergence analysis of the general Newton-type algorithm. The second aim of this thesis is a detailed discussion of these convergence assumptions for the case that the equation function of the constrained system is piecewise continuously differentiable. Some of the convergence assumptions seem quite technical and difficult to check. Therefore, we look for sufficient conditions which are still mild but which seem to be more familiar. We will particularly prove that the whole set of the convergence assumptions holds if some set of local error bound conditions is satisfied and in addition the feasible set of the constrained system excludes those zeros of the selection functions which are not zeros of the equation function itself, at least in a sufficiently small neighborhood of some fixed solution. We apply our results to constrained systems arising from complementarity systems, i.e., systems of equations and inequalities which contain complementarity constraints. Our new conditions are discussed for a suitable reformulation of the complementarity system as constrained system of equations by means of the minimum function. In particular, it will turn out that the whole set of the convergence assumptions is actually implied by some set of local error bound conditions. In addition, we provide a new constant rank condition implying the whole set of the convergence assumptions. Particularly, we provide adapted formulations of our new conditions for special classes of complementarity systems. We consider Karush-Kuhn-Tucker (KKT) systems arising from optimization problems, variational inequalities, or generalized Nash equilibrium problems (GNEPs) and Fritz-John (FJ) systems arising from GNEPs. Thus, we obtain for each problem class conditions which guarantee local quadratic convergence of the general Newton-type algorithm and its special realizations to a solution of the particular problem. Moreover, we prove for FJ systems of GNEPs that generically some full row rank condition is satisfied at any solution of the FJ system of a GNEP. The latter condition implies the whole set of the convergence assumptions if the functions which characterize the GNEP are sufficiently smooth. Finally, we describe an idea for a possible globalization of our Newton-type methods, at least for the case that the constrained system arises from a certain smooth reformulation of the KKT system of a GNEP. More precisely, a hybrid method is presented whose local part is the LP-Newton method. The hybrid method turns out to be, under appropriate conditions, both globally and locally quadratically convergent.

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Huang, Yun-ru, and 黃韻如. "Viscosity Approximation Methods for Generalized Equilibrium Problems and Fixed Point Problems." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/rrwzvc.

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碩士 國立中山大學 應用數學系研究所 96 The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a generalized equilibrium problem (for short, GEP) and the set of fixed points of a nonexpansive mapping in a Hilbert space. First, by using the well-known KKM technique we derive the existence and uniqueness of solutions of the auxiliary problems for the GEP. Second, on account of this result and Nadler''s theorem, we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the GEP and the set of fixed points of the nonexpansive mapping. Furthermore, it is proven that the sequences generated by this iterative scheme converge strongly to a common element of the set of solutions of the GEP and the set of fixed points of the nonexpansive mapping.

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Liu, Yu Hsuan, and 劉毓璿. "The Study of Abstract Economies, System of Generalized Vector Quasi-Equilibrium Problems and Optimization Problems." Thesis, 2003. http://ndltd.ncl.edu.tw/handle/83876535062785024795.

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碩士 國立彰化師範大學 數學系 91 In this paper, we apply Himmelberg''s fixed point theorem to establish existence theorems of equilibria for generalized abstract economies in which strategic spaces may not be compact and the set of players may not be countable. We apply our res- ults to establish general existence theorems of maximal elements and to establish existence theorems of system of generalized vector quasi-equilibrium problems from which we derive existence theorems of system of generalized vector quasi-variational and quasi-variational-like inequality problems and system of vector quasi-optimization problems.

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Chen, Li-fang, and 陳俐芳. "Existence Theorems of Abstract Economies and System of Generalized Vector Quasi-Equilibrium Problems with Applications." Thesis, 2003. http://ndltd.ncl.edu.tw/handle/26601269909427518675.

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碩士 國立彰化師範大學 數學系 91 In this paper, we first establish the existence theorems of generalized abstract economy with a lower semicontinuous constraint correspondence and a fuzzy constraint correspondence defined on non-compact and non-paracompact strategy sets. As consequence of our results, we get existence theorems for a solution to the system of generalized vector quasi-equilibrium problems. As applications, we derive some existence results for the system of mixed vector quasi-variational-like inequality problems, Debreu type vector equilibrium problems and Nash type equilibrium problem for vector-valued mappings.

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Peng, Bo-Jun, and 彭柏鈞. "Strong Convergence Theorems for Fixed Points and Systems of Generalized Equilibrium Problems with Applications to Optimization Theory." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/38402676636001742958.

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碩士 國立彰化師範大學 數學系所 99 In this paper, we find a common solution for the system of equilibrium problems and fixed points of a strictly pseudo-contractive mapping. We prove the strong convergence theorem based on shrinking projection in a real Hilbert space. Then we apply our results to study a common solution of system of mixed equilibrium problems and fixed point of a strict pseudo-contractive mapping in a Hilbert space mathematical program with system of mixed equilibrium problems and fixed points constraints.

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Liu, Xinwei, and Jie Sun. "Generalized Stationary Points and an Interior Point Method for MPEC." 2003. http://hdl.handle.net/1721.1/3701.

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Mathematical program with equilibrium constraints (MPEC)has extensive applications in practical areas such as traffic control, engineering design, and economic modeling. Some generalized stationary points of MPEC are studied to better describe the limiting points produced by interior point methods for MPEC.A primal-dual interior point method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced without assuming strict complementarity or linear independence constraint qualification. Under very general assumptions, the algorithm can always find some point with strong or weak stationarity. In particular, it is shown that every limiting point of the generated sequence is a piece-wise stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a certain point with weak stationarity can be obtained. Preliminary numerical results are satisfactory, which include a case analyzed by Leyffer for which the penalty interior point algorithm failed to find a stationary solution. Singapore-MIT Alliance (SMA)

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