# Seminar2022-2

Organized by Max L. N. Gonçalves

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The seminars in this semester will be held in the Lecture Room of IME/UFG, unless otherwise stated. All interested are very welcome to attend.

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**Date: ** November 03

**Speaker:** Max Leandro Nobre Gonçalves

**Title: **A Cubic Regularization of Newton Method with Finite-Difference Hessian Approximations

**Abstract: **In this talk, we present a version of the Cubic Regularization of the Newton method for unconstrained nonconvex optimization, in which the Hessian matrices are approximated by forward gradient differences. The regularization parameter of the cubic models and the accuracy of the Hessian approximations are jointly adjusted using a nonmonotone line-search criterion. Complexity analysis of the proposed algorithm is discussed and preliminary numerical experiments are presented to confirm our theoretical findings.

**Date: ** November 10

**Speaker:** Glaydston de Carvalho Bento

**Title: **Fenchel Conjugate via Busemann Function on Hadamard Manifolds

**Abstract: **In this paper we introduce a Fenchel-type conjugate, given as the supremum of convex functions, via the Busemann functions. Due the absence of non-constant affine functions in Hadamard manifolds and taking into account that the Busemann functions are smooth convex functions with constant gradient, our study ensures that our proposal of Fenchel conjugate seems to be the most adequate to cover the absence of approximations by non-constant affine functions. Besides, it is possible to evidence the influence of the sectional curvature in obtaining the main results. In particular, we have illustrated that the difference between a proper, lsc, convex function and its biconjugate is a constant that depends on the sectional curvature of the manifold, showing that in general a Fenchel-Moreau type theorem is directly influenced by the sectional curvature.

We also present some applications formulated in terms of the Fenchel's conjugate.

**Date:**November 17

**Speaker:** Erik Alex Papa Quiroz

**Title: **An inexact scalarization proximal point algorithm for constrained multiobjective minimization with quasiconvex objective functions

**Abstract:**In this talk, an inexact proximal point algorithm with proximal distances is introduced

objective functions constrained to a closed convex set with nonempty interior. We

prove the convergence of the sequence generated by the algorithm to a Pareto-Clarke

critical point and convergence to a Pareto solution when the objective functions are

convex or when we consider the exact proximal algorithm for the quasiconvex problem

when the proximal parameter converges to zero. We give some conditions to obtain

the finite termination of the algorithm and prove linear or superlinear rate of

convergence for a large class of proximal distances. Thus, this work extends the

convergence of the proximal point method for quasiconvex multiobjective minimization

problems and improves recent convergence results of previous works for the convex

case

**Date:**December 01

**Speaker:** Orizon P. Ferreira

**Title:**Copositivity with respect to general cones and existence results for complementarity problems

**Abstract:**We test copositivity of operators with respect to general cones by using a gradient projection algorithm for solving constrained convex problems on the sphere in finite dimensional vector spaces. This approach can also be used to analyse solvability of nonlinear cone-complementarity problems. The convergence analysis of the gradient projection algorithm on the sphere is also presented. To our best knowledge this is the first algorithm which can be used to test copositivity of operators with respect to general cones. Numerical results, including testing such copositivity of operators, are presented

**Date:**December 08

**Speaker:** Paulo César

**Title:**Convergence of Quasi-Newton Methods for Solving Constrained Generalized Equations

**Abstract:**In this paper, we focus on quasi-Newton methods to solve constrained generalized equations.

**Date:**January 12

**Speaker:** Luis Roman L. Perez

**Title:**Nonlinear conjugate gradient methods for Vector Optimization with approximated steepest descent direction and without Wolfe conditions

**Abstract:**We will present three variants of our algorithm. The first is for convex problems, the second uses the Lipschitz constant of the objective, and finally, a variant that needs a good approximation of the Lipschitz constant.

**Date:**January 19

**Speaker:** Jefferson D. G. Melo

**Title:**Proximal Gradient Methods for Composite Multiobjective Optimization

**Abstract:**In this talk, we will review some proximal gradient methods for solving composite multiobjective/vectorial optimization problems whose objective function can be expressed as the sum of two convex functions, a differentiable and a prox-friendly non-smooth one. We will also discuss two line-searches procedures that allow us to consider problems for which the Lipschitz constant of the Jacobian of the differentiable component of the objective function is hard to compute or does not exist.

**Date:**February 02

**Speaker:** Danilo R. Souza

**Title:**Métodos Quase-Newton com Busca Linear de Wolfe para Otimização Multiobjetivo.

**Abstract:**Propomos três métodos tipo BFGS com busca linear de Wolfe para otimização multiobjetivo irrestrita. Os algoritmos são bem definidos mesmo para problemas gerais não convexos. O primeiro mimetiza o método BFGS clássico para otimização escalar, para o qual a convergência global e R-linear para um ponto Pareto ótimo são estabelecidas para problemas fortemente convexos. Na análise de convergência local, a taxa é Q-superlinear. Os outros dois algoritmos são versões globalmente convergentes do método BFGS para problemas não convexos. Finalmente, caracterizamos explicitamente de maneira não assintótica a convergência local superlinear do método BFGS para otimização multiobjetivo.

**Date:**February 09

**Speaker:** Paulo César

**Title:**Globally convergent inexact quasi-Newton methods for

**Abstract:**Large scale nonlinear systems of equations can be solved by means of inexact

**Schedule:**

10 - Glaydston

17 - Erick

24 - Conpeex

December

1 - Orizon

08 - Paulo César

15 - Leandro

January

12 - Luis Roman

19 - Jefferson

26 - Paulo César

Fevereiro

2 - Danilo

9 -