## Seminar 2016.1

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Organized by  Jefferson D. G. Melo
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The seminars in this semester will be held  in the Lecture Room  of  IME/UFG, unless otherwise stated. All who are interested in  are very welcome.

08/04,  16:00 - 17:00
Max L.N. Goncalves (Professor IME/UFG)
A Newton conditional gradient method for constrained nonlinear systems
Abstract:In this talk, we will consider the problem of solving a constrained system of nonlinear equations. We will propose an algorithm based on a combination of the Newton and conditional gradient methods, and establish its local convergence analysis. Our analysis will be set up by using a  majorant condition technique, allowing us to prove in a unified way convergence results for  two large families of nonlinear functions.  The first one includes functions whose derivative satisfies a Holder-like condition, and the second one consists of a substantial subclass of analytic functions.  Numerical experiments illustrating the applicability of the proposed method will be discussed. This is a joint work with Jefferson G. Melo.

15/04, 16:00-17:00
Gilson do Nascimento Silva (Phd Student IME/UFG)
Unifying  the local convergence analysis  of Newton's  Method for strongly regular generalized equations
Abstract: In this talk we consider Newton's method for solving the generalized equation in Banach spaces of the type 0 \in  f(x) + F(x), where f is a differentiable function and F is a set-valued mapping with closed graph. We show, under strong regularity of the generalized equation (concept introduced by S.M.Robinson in 1980), that the Newton's method is local  quadratically convergent to a solution.  Our approach is based on the Banach Perturbation Lemma for generalized equation and the majorant condition, relaxing the Lipschitz continuity of the derivative f'. We obtain  the optimal convergence radius,  uniqueness of  solution and unify some convergence results in the related Literature.

29/04, 16:00-17:00
Yuri Rafael L. Pereira (Phd Student IME/UFG)
Trust-Region Method for Unconstrained Mutiobjective Problems on Riemannian Manifolds
Abstract: In this talk, we extend the  multiobjective version  of the trust-region method to the Riemannian context.  The method consists of monitoring the agreement between  the  objective function and the quadractic model  along some descent directions, computed at the starting point. Under certain assumptions,  we prove that the  trust-region method generates a sequence which converges to a Pareto critical  point.

06/05,  13:15-14:45
Edvaldo E. A. Batista (Phd Student IME/UFG)
Abstract: Neste algoritmo extragradiente  estendemos, para campos vetoriais em variedades de Hadamard, métodos desenvolvidos em espaços Euclidianos para solucionar o VIP(T;C), onde T é um operador ponto-conjunto e C é um conjunto convexo e fechado. Além disto, estendemos à Variedades de Hadamard a importante semicontinuidade inferior do operador $T^{\epsilon}$.

13/05, 13:15-14:45
Gilson do Nascimento Silva (Phd Student IME/UFG)
Kantorovich's theorem on Newton's method for solving strongly regular generalized equation
Abstract: In this talk we consider the Newton's method for solving the generalized equation of the form  f(x) +F(x) \ni 0, where f:X \to Y is a   continuously   differentiable mapping, X  and Y are Banach spaces, and F:X \rightrightarrows  Y be a set-valued mapping with  nonempty  closed graph.   We show that, under strong regularity of the generalized equation,  concept introduced by S.M.Robinson in  1980, and starting point satisfying the  Kantorovich's assumptions,  the Newton's method is quadratically convergent to a solution, which is unique in a suitable neighborhood  of the starting point.  The analysis presented   based on Banach Perturbation Lemma for generalized equation and the majorant technique, allow  to  unify some results  pertaining the Newton's method theory.

20/05, 13:15-14:45
Jose Yunier Bello Cruz  (Professor IME/UFG)
On Forward-Backward Splitting Iteration for Inclusion Problems
Abstract: In this talk, we reveal many hidden ideas of the forward-backward splitting method for finding a zero of the sum of two maximal monotone mappings. It is worth mentioning that standard convergence and complexity analyses of the (one step) forward-backward iteration requires at least a co-coercivity assumption of one (point-to-point) operator and the stepsizes to lie within a suitable interval related with the co-coercivity constant. It is well-known that co-coercive operators are monotone and Lipschitz continuous, but the converse does not hold in general. Although, for gradients of lower semicontinuous, proper and convex functions, the co-coercivity is equivalent to the global Lipschitz continuity assumption (Baillon-Haddad Theorem). This is a nice and surprising fact, which is strongly used in the convergence analysis of the proximal forward-backward method for minimizing the sum of two convex functions. Here, we focus on Tseng scheme and some interesting possible modifications.

The content of this talk is based on the following papers:

[1] J.Y. Bello Cruz, On proximal subgradient splitting method for minimizing the sum of two nonsmooth convex functions. Available in ArXiv 20 p.,
[2] J.Y. Bello Cruz, R. Díaz Millán, A variant of forward-backward splitting method for the sum of two monotone operators with a new search strategyOptimization 64 (2015), pp.1471--1486.
[3] P. Neal, S. Boyd, Proximal Algorithms, Foundations and Trends in Optimization 1 (2014), pp. 127--239.
[4] R.T. Rockafellar, Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization 14 (1976), pp. 877--898.
[5] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM Journal on Control Optimization 38 (2000), pp. 431--446.

03/06, 13:15-14:45
Edvaldo E. A. Batista (Phd Student IME/UFG)
A Korpolevich Method for Variational Inequalities Problems in Hadamard Manifolds
Abstract: We propose a Korpolevich method for solving the variational inequality problem in Hadamard manifolds, where the underlying vector field is continuous and satisfies a certain generalized monotonicity assumption (e.g., it can be pseudomonotone). The method is simple and admits a nice geometric interpretation. It consists of two steps. First, we construct an appropriate "hyperplane" which strictly separates the current iterate from the solutions of the problem. This procedure requires a single projection onto the feasible set and employs an Armijo-type linesearch along a feasible direction. Then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the "halfspace" containing the solution set. Thus, in contrast with most other projection-type methods, only two projection operations per iteration are needed.

17/06, 13:15-14:45
Jefferson G. Melo (Professor IME/UFG)
complexidade de metodos de primeira ordem
Resumo: Neste seminario discutiremos sobre complexidade de metodos de primeira ordem, dando enfase nos metodos do gradiente, ponto proximal e metodos acelerados de Nesterov. Discutiremos sobre a otimalidade dos  metodos de Nesterov  e apresentaremos uma prova desse resultado para uma variante dos metodos acelerados de Nesterov.

24/06,  13:15-14:45
Lucas (Phd Student IME/UFG)
Nova Abordagem do Algoritmo do Ponto Proximal no contexto Riemanniano
Resumo: Nesta exposição será apresentado o método do ponto proximal clássico para problemas de minimização no contexto Euclidiano e sua extensão ao contexto Riemanniano introduzido por Ferreira e Oliveira [1]. Neste seminario daremos enfase  à aplicabilidade do método sobre variedades compactas e com sinal da curvatura seccional não constante. Para exemplos de tais variedades e motivações, ver Rapcsák [2] e  Absil [3], onde autores trazem exemplos de variedades de curvatura não constante e motivação para estudar  em tais ambientes. Tambem apresentaremos uma análise de convergência do método, segundo a abordagem dada por Bento, Cruz Neto e Oliveira [4], para funções que satisfazem a desigualdade de Kurdyka-Lojasiewicz.

Referencias:
[1]  Ferreira, O.P., Oliveira,P.R: Proximal point algorithm on Riemannian Manifold. Optimization 51, 257-270 (2002).
[2] Rapcsák, T.: Sectional curvature in nonlinear  optimization. J.Glob.Optim. 40(1-3), 375-388 (2008).
[3] Absil, P. A., Mahony, R., Sepulchre, R.:Optimization on Manifolds: Methods and Applications. Recent Advances in Optimization and its Applications in Engineering. Springer, New York (2010).
[4] Bento, G. C., Cruz Neto, J. X., Oliveira, P. R.: A New Approach of Proximal Point Method on Riemannian Manifolds.JOTA (2015)

24/06, 13:15-14:45
Fabiana Rodrigues de Oliveira (Phd Student IME/UFG)
Local convergence analysis of a proximal Gauss-Newton method under a majorant condition
Abstract: The goal of this seminar is to present the main ideas of the paper "Local convergence analysis of a proximal Gauss-Newton method under a majorant condition",  authors Max L.N. Gonçalves and Gemayqzel B. Allende. It will be presented the proximal Gauss-Newton method for solving penalized nonlinear least squares problems. A local convergence analysis will be  obtained under the assumption that the derivative of the function associated with the penalized least square problem satisfies a majorant condition.

01/07, 13:15-14:45
Fabrícia Rodrigues de Oliveira (Phd Student IME/UFG)
Método de região de confiança para sistemas de equações não-lineares com restrições de caixa
Resumo: Será apresentado um método iterativo para resolver sistemas de equações não-lineares com restrições de caixa. Este método generaliza a ideia de região de confiança para sistemas de equações não-lineares irrestrito para o problema com restrições de caixa.

15/07, 13:15-14:45