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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.
| Identifer | oai:union.ndltd.org:IBICT/oai:repositorio.bc.ufg.br:tede/4562 |
| Date | 27 February 2015 |
| Creators | Millán, Reinier Díaz |
| Contributors | Cruz, José Yunier Bello Cruz, Cruz, José Yunier Bello, Iusem, Alfredo Noel, Makler, Susana Scheimberg de, Sandoval, Wilfredo Sosa, Ferreira, Orizon Pereira |
| Publisher | Universidade Federal de Goiás, Programa de Pós-graduação em Matemática (IME), UFG, Brasil, Instituto de Matemática e Estatística - IME (RG) |
| Source Sets | IBICT Brazilian ETDs |
| Language | Portuguese |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, info:eu-repo/semantics/doctoralThesis |
| Format | application/pdf |
| Source | reponame:Biblioteca Digital de Teses e Dissertações da UFG, instname:Universidade Federal de Goiás, instacron:UFG |
| Rights | http://creativecommons.org/licenses/by-nc-nd/4.0/, info:eu-repo/semantics/openAccess |
| Relation | 6600717948137941247, 600, 600, 600, 600, -4268777512335152015, -7090823417984401694, 2075167498588264571 |
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