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.
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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.
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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
to solve multiobjective minimization problems with locally Lipschitz quasiconvex
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
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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
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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.
As is well-known, this problem was firstly studied by Robinson and Josephy in the 70's.
Since then, it has been extensively studied by many other researchers, specially Dontchev and Rockafellar.
Here, we propose a Broyden-type quasi-Newton approach to deal with constrained generalized equations. Projections onto the feasible set are inexact. The local convergence of general quasi-Newton approach is established under a bounded deterioration of the update matrix and Lipschitz continuity hypotheses. In particular, we prove that a general scheme converges linearly to the solution under suitable assumptions.
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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.
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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.
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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.
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Title: Globally convergent inexact quasi-Newton methods for
solving nonlinear systems
Abstract: Large scale nonlinear systems of equations can be solved by means of inexact
quasi-Newton methods. A global convergence theory is introduced that guarantees
that, under reasonable assumptions, the algorithmic sequence converges to a solution
of the problem. Under additional standard assumptions, superlinear convergence is
preserved.
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Schedule:
November
3 - Max
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 -
16-
