Invited Session Tue.1.H 2051

Tuesday, 10:30 - 12:00 h, Room: H 2051

Cluster 24: Variational analysis [...]

Variational inequalities and optimization problems on Riemannian manifolds

 

Chair: Genaro López and Chong Li

 

 

Tuesday, 10:30 - 10:55 h, Room: H 2051, Talk 1

Vittorio Colao
Equilibrium problems in Hadamard manifolds

 

Abstract:
Equilibrium problems in linear spaces had been widely investigated
in recent years and by several authors. It had been proved that a
broad class of problems, such as variational inequality, convex minimization,
fixed point and Nash equilibrium problems can be formulated as equilibrium
problems.
In this talk, I will deal with equilibrium problems in in the setting
of manifolds with nonpositive sectional curvature. An existence result
will be presented, together with applications to variational inequality,
fixed point for multivalued maps and Nash equilibrium problems. I
will also introduce a firmly nonexpansive resolvent and discuss an
approximation result for equilibrium points.

 

 

Tuesday, 11:00 - 11:25 h, Room: H 2051, Talk 2

Laurentiu Leustean
Firmly nonexpansive mappings in classes of geodesic spaces

Coauthors: David Ariza-Ruiz, Genaro Lopez-Acedo

 

Abstract:
Firmly nonexpansive mappings play an important role in metric fixed point theory and optimization due to their correspondence with maximal monotone operators. In this paper we do a thorough study of fixed point theory and the asymptotic behaviour of Picard iterates of these mappings in different classes of geodesic spaces, as (uniformly convex) W-hyperbolic spaces, Busemann spaces and CAT(0) spaces. Furthermore, we apply methods of proof mining to obtain effective rates of asymptotic regularity for the Picard iterations.

 

 

Tuesday, 11:30 - 11:55 h, Room: H 2051, Talk 3

Paulo Roberto Oliveira
Proximal and descent methods on Riemannian manifolds

Coauthors: Glaydston Carvalho Bento, João Xavier Cruz Neto, Erik Alex Papa Quiroz

 

Abstract:
This talk has two parts. In the first, it is analyzed the proximal point method applied in Hadamard manifolds, associated to the corresponding distance. The considered functions are locally Lypschitz quasiconvex. Under reasonable hypothesis, it is proved the global convergence of the sequence generated by the method to a critical point. In the second part, the concerned class are lower semicontinuous functions satisfying Kurdyka-Lojasiewicz property . An abstract convergence analysis for inexact methods in Riemannian manifolds allows to obtain full convergence of bounded sequences applied to proximal method, associated to a quasi-distance (the usual distance without symmetry). The results are independent of the curvature of the manifold. A second application of the abstract theory is the convergence of inexact descent method for that class of functions on Hadamard manifolds. This extends known results for Riemannian manifolds with positive curvature. Finally, some applications are cited in related papers.

 

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