Seminar 2014.2

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Organized by   Max Leandro Nobre Gonçalves
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13/08/2014, 14:00 -15:00
Sándor Zoltán Németh (Professor da school of mathematics, the university of Birmingham, Birmingham, UK)       
Lattice-like sets and isotone projections. Theoretical and practical applications
Abstract: While studying some properties of linear operators in a Euclidean Jordan algebra, Gowda, Sznajder and Tao have introduced generalized lattice operations based on the projection onto a self-dual cone and in particular onto the cone of squares. We have extended these operations to more general cones.

We have shown that these lattice-like operations and their generalizations are important tools in establishing the isotonicity of the metric projection onto some closed convex sets. The results of this kind are motivated by methods for proving the existence of solutions of variational inequalities (and in particular complementarity problems) and methods for finding these solutions in a recursive way.

It turns out, that the closed convex sets admitting isotone (i.e., order preserving) projections are exactly the sets which are invariant with respect to these lattice-like operations, called lattice-like sets. A nice theoretical application of this property is to show that the projection onto a closed convex cone is isotone with respect to the cone (i.e., the order defined by the cone) if and only if the projection onto the dual of the cone is subadditive with respect to the dual cone (i.e., the order defined by the the dual cone).

For the nonnegative orthant, the Lorentz cone, and extended Lorentz cones we determined the lattice-like sets.
We have given recursive methods to solve generalized mixed complementarity problems, by using the isotonicity of the projection with respect to an extended Lorentz cone. We have given conditions for a set to be lattice-like with respect to a simplicial cone.

We have considered the problem of lattice-like sets in Euclidean Jordan algebras with respect to the cone of squares. We have shown that the Jordan subalgebras are lattice-like sets, but the converse in general is not true. In the case of simple Euclidean Jordan algebras of rank at least three the lattice-like property is rather restrictive, e.g., there are no lattice-like proper closed convex sets with interior points. The case of  simple Euclidean Jordan algebras of rank two is equivalent to determining the lattice-like sets with respect to a Lorentz cone

20/08/2014, 14:00 -15:00
Max Leandro N. Gonçalves (Professor IME/UFG)       
Local convergence analysis of a proximal Gauss-Newton method under a majorant condition
Abstract: In this talk, the proximal Gauss-Newton method for solving penalized nonlinear least squares problems is studied.  A local convergence analysis is obtained under the assumption that the derivative of the function associated with the penalized least square problem satisfies a majorant condition. Our analysis provides a clear relationship between the majorant function and the  function associated with the penalized least squares problem. The convergence for two important special cases is also derived.

29/08/2014, 13:00-14:00
Orizon Pereira Ferreira (Professor IME/UFG)
Projection onto simplicial cones by a semi-smooth Newton method
Abstract: In this paper the problem of projection onto a  simplicial cone  is studied. By using Moreau's decomposition theorem for projecting onto closed convex cones, the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations. It is shown that a semi-smooth Newton method applied to the  system of equations associated  to the  problem of projecting onto a simplicial cone is always  well defined, and  the generated sequence is   bounded  for any starting point and a formula for any accumulation point of this sequence is presented. It is also shown that   under a somewhat restrictive assumption, the semi-smooth Newton method applied to the system of equations associated to the problem of projecting onto a simplicial cone has finite convergence. Besides, under a mild assumption on the simplicial cone, the generated sequence converges linearly to the solution of the associated system of equations.

05/09/2014, 13:00-14:00
Gilson do Nascimento Silva, (Phd Student IME/UFG)
Backward-backward splitting em espaços de Hadamard
Abstract: O algoritmo backward-backward splitting é uma ferramenta para encontrar mínimo de regularização da soma de duas funções convexas em espaços de Hilbert. Neste trabalho será apresentado uma generalização para espaços de Hadamard (espaços CAT(0) ) e uma prova de convergência do método backward-backward sob um erro no cálculo do ponto proximal.

12/09/2014, 13:00-14:00
Edvaldo E. A. Batista  (Phd Student IME/UFG)
Um resultado de existência para o problema de quase-equilíbrio vetorial generalizado em variedades de Hadamard
Resumo: Considerando o PQEVG em variedades de Hadamard, utilizaremos uma versão em variedades do teorema de KKM para estender um resultado de existência do PQEVG em espaços vetoriais topológicos para o contexto Riemanniano.

26/09/2014, 13:00-14:00
Luis Roman Lucambio Pérez  (Professor IME/UFG)
Non-linear conjugate gradients algorithm in vector optimization
Abstract:  In this talk, a new algorithm for solving vector optimization problems without restrictions is proposed. This algorithm will be an extension of the well known non-linear conjugate gradients method to vector optimization. We will formulate this algorithm with standard Wolfe condition in the line search. We will show that the algorithm is of descent-direction-type, that, assuming that at each iteration, the line search with standard Wolfe condition is successfull, that the Jacobian is Lipschitzian and that a condition of below boundedness is valid, then a property, analogous to the Zoutendijk's condition, is true, and therefore, the algorithm has global convergence properties.

03/10/2014, 13:00-14:00
Glaydston Carvalho Bento (Professor IME/UFG)
A generalized inexact proximal point method for nonsmooth functions that satisfy Kurdyka Lojasiewicz inequality
Abstract:  In this talk, following the ideas presented in Attouch et al. (Math. Program. Ser. A, 137: 91-129, 2013), we present an inexact version of theproximal point method for nonsmoth functions, whose regularization is givenby a generalized perturbation term. More precisely, the new perturbationterm is defined as a \textquotedblleft curved enough" function of the quasidistance between two successive iterates, that appears to be a nice tool for Behavioral Sciences (Psychology, Economics, Management, Game theory, \ldots). Our convergence analysis is a extension, of the analysis due to Attouch and Bolte (Math. Program. Ser. B, 116: 5-16, 2009) or, more generally, to Moreno et al. (Optimization, 61:1383-1403, 2012), to an inexact setting of the proximal method which is more suitable from the point of view of applications.

10/10/2014, 13:00-14:00
Valdinês Leite de Sousa Júnior, (Phd Student IME/UFG)
Um Método Abstrato de Descida Inexato para uma classe especial de funções não convexas
Resumo:  Neste seminário analisamos a convergência de um método abstrato que, sob certas condições, atinge um ponto crítico de um problema de minimização. O método (resp. a análise de convergência) apresentados são fruto de pesquisa atual que generalizam o método abstrato (resp. a análise de convergência) considerado em [1].

[1] Alaa Eddine, N., Pierre, M. Convergence to equilibrium for discretized gradient-like
systems with analytic features, IMA J. Numer. Anal. 33 (2013), pp. 1291?1321.

17/10/2014, 13:00-14:00
Alina Ruiz (professora da Universidad de la Habana, Cuba)    
On the topology of global optimization
Abstract:  We succintly describe the basic idea of Morse theoryin finite-dimensional smooth optimization. This is concerned with critical points (in particular, Karush?Kuhn?Tucker points) and relations between them.Then, we turn to gradient flows and focus on the fundamental problem: how to get from one local minimum to (all) other ones.

31/10/2014, 13:00-14:00
Jorge Barrios Ginart(Aluno de pós-doutorado do IME/UFG)
Projection onto simplicial cones by Picard's method. Numerical experiments
Abstract:  By using Moreau's decomposition theorem for projecting onto cones, the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations. It is shown that  the  Picard's method applied to the  system of equations associated  to the  problem of projecting onto a simplicial cone  generate  a sequence  that converges linearly to the  solution of the  system. Numerical experiments are presented making the comparison between  Picard's and semi-smooth Newton's methods to solve the nonsmooth system associated with the problem of projecting a point onto a simplicial cone.

07/11/2014, 13:00-14:00
Reinier Diaz Millan, (Phd Student IME/UFG)  
Um algoritmo extragradiente para o problema de desigualdade variacional sem monotonia
Resumo:  Nesta apresentação,  proporemos um método extragradiente onde retiramos toda hipóteses de  monotonia sobre o operador ponto-ponto. Somente exigiremos que o operador seja continuo e que o problema dual tenha solução.

14/11/2014, 13:00-14:00
Ademir  Aguiar, (Aluno de mestrado do IME/UFG)
Convergência Semi-Local do Método de Gauss-Newton sob uma condição majorante
Resumo:  Neste seminário apresentaremos uma análise semi-local de convergência local do método de Gauss-Newton para uma classe especial de sistemas de equações não-lineares. Sob a hipótese de que a derivada do operador não linear em consideração satisfaz uma condição majorante. Identificamos regiões, onde para o problema em consideração, o método de Gauss-Newton está bem-definido. Também apresentaremos casos especiais da teoria geral como aplicações.

21/11/2014, 13:00-14:00
Sira Ma. Allende Alonso ( Professora da Universidad de la Habana, Cuba)
Heuristic and hybrid algorithm for global optimization problem on discrete set
Resumo:  Numerous decision situations are modeled as Optimization problems on discrete sets.  Computational complexity of many of such models does not allow the use of exact algorithm for solving them. Heuristic methods emerge in those cases as a via of solution. Metaheuristics offer a framework for designing heuristic algorithms. Concepts themes and tools of this important and evolving area of optimization will be discussed in the presentation. They will be also shown in the specific contexts of applications. Finally, some ideas for tackling continuous problems using metaheuristics are proposed.

28/11/2014, 13:00-14:00
Vando Adona, (Aluno de mestrado do IME/UFG)
Método Subgradiente Incremental para Otimização Convexa não Diferenciável
Resumo : Consideramos um problema de minimização restrito, cuja função  objetivo  consiste na soma de funções convexas,  não necessariamente diferenciáveis. Estudamos um método subgradiente que executa a iteração de forma incremental, selecionando cada função componente de maneira sequencial e processando a iteração subgradiente individualmente. Analisamos diferentes alternativas para a escolha do comprimento de passo,  destacando as propriedades de convergência para cada caso.

05/12/2014, 13:00-14:00
Pedro, (Aluno de mestrado do IME/UFG)
Um Algorítmo Proximal com Quase-distância
Resumo: Estudaremos a convergência do método do ponto proximal (M.P.P), regularizado por uma quase distância, aplicado a um problema de otimização irrestrita. A função objetivo considerada não é necassariamente convexa e satisfaz a propriedade de Kurdyka-Lojasiewicz ao redor de seus pontos críticos generalizados. Mais precisamente, mostramos que qualquer sequência limitada gerada pelo (M.P.P), converge a um ponto crítico generalizado.