Cette thèse relate certains aspects liés à l'analyse post-Pareto issue de Problèmes d'Optimisation Vectorielle Stochastique. Un problème d'optimisation Vectorielle Stochastique consiste à optimiser l'espérance d'une fonction vectorielle aléatoire définie sur un ensemble arbitraire et à valeurs dans un espace sectoriel ordonné. L'ensemble des solutions de ce problème (appelé ensemble de Pareto) est composé des solutions admissibles qui assurent un certain équilibre entre les objectifs : il est impossible d'améliorer la valeur d'un objectif sans détériorer celle d'un autre. D'un point de vue technique, chaque solution de Pareto est acceptable. Nous nous posons alors le problème de la sélection de l'une d'entre elles : en supposant l'existence d'un décideur qui aurait son propre critère de décision, nous considérons le problème post-Pareto Stochastique qui vise à minimiser cette fonctionnelle sur l'ensemble de Pareto associé à un Problème d'Optimisation Vectorielle Stochastique. / This thesis explore related aspects to post-Pareto analysis arising from Stochastic Vector Optimization Problem. A Stochastic Vector Optimization Problem is to optimize a random vector objective function defined on an arbitrary set, and taking values in a partially ordered set. Its solution set (called Pareto set) consists of the feasible solutions which ensure some sort of equilibrium amongst the objectives. That is to say, Pareto solutions are such that noneof the objectives values can be improved further without deterioring another. Technically speaking, each Pareto solution is acceptable. The natural question that arises is : how to choose one solution ? One possible answer is to optimize an other objective over the Pareto set. Considering the existence of a decision-maker with its own criteria, we deal with the post-Pareto Stochastic Optimization Problem of minimizing its real-valued criteria over the Pareto set.

Many professions occasionally involve the selection of an alternative from among many problem solutions which have impacts in multiple-interest areas; however, due to the very nature of his work, the practicing engineer, regardless of specialty, is unavoidably engaged in this selection process. The emergence of national concern for environmental and social consequences of technical enterprises, as reflected through legislative action, has accentuated the need for multicriteria design methodologies in some areas of engineering (i.e., automotive). Consequently, interest in the development of pragmatic and theoretically sound approaches to multi-impact design situations has been keen. Any approach to multicriteria design/decision problems involves two fundamental aspects: (1) generating information regarding the range of possible designs and their associated impacts; and (2) generating relative value information which is used to compare the relative imp-cats leading to the selection of a "preferred" or "best compromise" alternative. The methodology developed herein is the integration of a formal mathematical programming technique for generating the full range of feasible alternatives with a pragmatic and well-accepted group-interaction technique for extracting value information regarding alternatives. The integration results in an iterative group-interaction process which leads to successive reductions in the preferred range of alternatives until the most preferred alternative is identified. The methodology developed in this research represents an improvement over other methodologies reported in the literature in two areas: 1) The noninferior set is explicitly identified insuring selection of a group decision point which is noninferior, 2) a least squared error mathematical filtering technique is developed for smoothing relative value data obtained from the decision making body. In addition, a convergence proof is developed which not only indicates the theoretically sound and robust nature of the algorithm developed in this work but in addition provides a basis for an improved class of algorithms for solving classical nonlinear constrained problems. The technique was developed for and implemented in an interactive software package. The multiobjective decision problem is solved in a single encounter with a cooperative decision making group.

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.

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2017-09-21T21:10:08Z No. of bitstreams: 2 Tese - Yuri Rafael Leite Pereira - 2017.pdf: 2066899 bytes, checksum: e1bbe4df9a2a43e1074b83920a833ced (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-09-22T11:44:33Z (GMT) No. of bitstreams: 2 Tese - Yuri Rafael Leite Pereira - 2017.pdf: 2066899 bytes, checksum: e1bbe4df9a2a43e1074b83920a833ced (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-09-22T11:44:33Z (GMT). No. of bitstreams: 2 Tese - Yuri Rafael Leite Pereira - 2017.pdf: 2066899 bytes, checksum: e1bbe4df9a2a43e1074b83920a833ced (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-08-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we will analyze three types of method to solve vector optimization problems in different types of context. First, we will present the trust region method for multiobjective optimization in the Riemannian context, which retrieves the classical trust region method for minimizing scalar functions. Under mild assumptions, we will show that each accumulation point of the generated sequences by the method, if any, is Pareto critical. Next, the proximal point method for vector optimization and its inexact version will be extended from Euclidean space to the Riemannian context. Under suitable assumptions on the objective function, the well-definedness of the methods will be established. Besides, the convergence of any generated sequence, to a weak efficient point, will be obtained. The last method to be investigated is the Newton method to solve vector optimization problem with respect to variable ordering structure. Variable ordering structures are set-valued map with cone values that to each element associates an ordering. In this analyze we will prove the convergence of the sequence generated by the algorithm of Newton method and, moreover, we also will obtain the rate of convergence under variable ordering structures satisfying mild hypothesis. / Neste trabalho, analisaremos três tipos de métodos para resolver problemas de otimização vetorial em diferentes tipos contextos. Primeiro, apresentaremos o método da Região de Confiança para resolver problemas multiobjetivo no contexto Riemanniano, o qual recupera o método da Região de Confiança clássica para minimizar funções escalares. Sob determinadas suposições, mostraremos que cada ponto de acumulação das sequências geradas pelo método, se houver, é Pareto crítico. Em seguida, o método do ponto proximal para otimização vetorial e sua versão inexata serão estendidos do espaço Euclidiano para o contexto Riemanniano. Sob adequados pressupostos sobre a função objetiva, a boas definições dos métodos serão estabelecidos. Além disso, a convergência de qualquer sequência gerada, para um ponto fracamente eficiente, é obtida. O último método a ser investigado é o método de Newton para resolver o problema de otimização vetorial com respeito a estruturas de ordem variável. Estruturas de ordem variável são aplicações ponto-conjunto cujas imagens são cones que para cada elemento associa uma ordem. Nesta análise, provaremos a convergência da sequência gerada pelo algoritmo do método de Newton e, além disso, também obteremos a taxa de convergência sob estruturas de ordem variável satisfazendo adequadas hipóteses.

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.

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.