This work is devoted to study efficient numerical methods for solving nonsmooth convex equilibrium problems in the sense of Blum and Oettli. First we consider the auxiliary problem principle which is a generalization to equilibrium problems of the classical proximal point method for solving convex minimization problems. This method is based on a fixed point property. To make the algorithm implementable we introduce the concept of $mu$-approximation and we prove that the convergence of the algorithm is preserved when in the subproblems the nonsmooth convex functions are replaced by $mu$-approximations. Then we explain how to construct $mu$-approximations using the bundle concept and we report some numerical results to show the efficiency of the algorithm. In a second part, we suggest to use a barrier function method for solving the subproblems of the previous method. We obtain an interior proximal point algorithm that we apply first for solving nonsmooth convex minimization problems and then for solving equilibrium problems. In particular, two interior extragradient algorithms are studied and compared on some test problems.

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No. of bitstreams: 2 Tese - Reinier Díaz Millán - 2015.pdf: 3568052 bytes, checksum: b4c892f77911a368e1b8f629afb5e66e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-02-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Nesta tese apresentamos v arios algoritmos para resolver os problemas de Desigualdade Variacional e Inclus~ao. Para o problema de desigualdade variacional propomos, no Cap tulo 2 uma generaliza c~ao do algoritmo cl assico extragradiente, utilizando vetores normais n~ao nulos do conjunto vi avel. Em particular, dois algoritmos conceituais s~ao propostos e cada um deles cont^em tr^es variantes diferentes de proje c~ao que est~ao relacionadas com algoritmos extragradientes modi cados. Duas buscas diferentes s~ao propostas, uma sobre a borda do conjunto vi avel e a outra ao longo das dire c~oes vi aveis. Cada algoritmo conceitual tem uma estrat egia diferente de busca e tr^es formas de proje c~ao especiais, gerando tr^es sequ^encias com diferente e interessantes propriedades. E feito a an alise da converg^encia de ambos os algoritmos conceituais, pressupondo a exist^encia de solu c~oes, continuidade do operador e uma condi c~ao mais fraca do que pseudomonotonia. No Cap tulo 4, n os introduzimos um algoritmo direto de divis~ao para o problema variacional em espa cos de Hilbert. J a no Cap tulo 5, propomos um algoritmo de proje c~ao relaxada em Espa cos de Hilbert para a soma de m operadores mon otonos maximais ponto-conjunto, onde o conjunto vi avel do problema de desigualdade variacional e dado por uma fun c~ao n~ao suave e convexa. Neste caso, as proje c~oes ortogonais ao conjunto vi avel s~ao substitu das por proje c~oes em hiperplanos que separam a solu c~ao da itera c~ao atual. Cada itera c~ao do m etodo proposto consiste em proje c~oes simples de tipo subgradientes, que n~ao exige a solu c~ao de subproblemas n~ao triviais, utilizando apenas os operadores individuais, explorando assim a estrutura do problema. Para o problema de Inclus~ao, propomos variantes do m etodo de divis~ao de forward-backward para achar um zero da soma de dois operadores, a qual e a modi ca c~ao cl assica do forwardbackward proposta por Tseng. Um algoritmo conceitual e proposto para melhorar o apresentado por Tseng em alguns pontos. Nossa abordagem cont em, primeramente, uma busca linear tipo Armijo expl cita no esp rito dos m etodos tipo extragradientes para desigualdades variacionais. Durante o processo iterativo, a busca linear realiza apenas um c alculo do operador forward-backward em cada tentativa de achar o tamanho do passo. Isto proporciona uma consider avel vantagem computacional pois o operador forward-backward e computacionalmente caro. A segunda parte do esquema consiste em diferentes tipos de proje c~oes, gerando sequ^encias com caracter sticas diferentes. / In this thesis we present various algorithms to solve the Variational Inequality and Inclusion Problems. For the variational inequality problem we propose, in Chapter 2, a generalization of the classical extragradient algorithm by utilizing non-null normal vectors of the feasible set. In particular, two conceptual algorithms are proposed and each of them has three di erent projection variants which are related to modi ed extragradient algorithms. Two di erent linesearches, one on the boundary of the feasible set and the other one along the feasible direction, are proposed. Each conceptual algorithm has a di erent linesearch strategy and three special projection steps, generating sequences with di erent and interesting features. Convergence analysis of both conceptual algorithms are established, assuming existence of solutions, continuity and a weaker condition than pseudomonotonicity on the operator. In Chapter 4 we introduce a direct splitting method for solving the variational inequality problem for the sum of two maximal monotone operators in Hilbert space. In Chapter 5, for the same problem, a relaxed-projection splitting algorithm in Hilbert spaces for the sum of m nonsmooth maximal monotone operators is proposed, where the feasible set of the variational inequality problem is de ned by a nonlinear and nonsmooth continuous convex function inequality. In this case, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each iteration of the proposed method consists of simple subgradient-like steps, which does not demand the solution of a nontrivial subproblem, using only individual operators, which explores the structure of the problem. For the Inclusion Problem, in Chapter 3, we propose variants of forward-backward splitting method for nding a zero of the sum of two operators, which is a modi cation of the classical forward-backward method proposed by Tseng. The conceptual algorithm proposed here improves Tseng's method in many instances. Our approach contains rstly an explicit Armijo-type line search in the spirit of the extragradient-like methods for variational inequalities. During the iterative process, the line search performs only one calculation of the forward-backward operator in each tentative for nding the step size. This achieves a considerable computational saving when the forward-backward operator is computationally expensive. The second part of the scheme consists of special projection steps bringing several variants.