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Instituto Alberto Luiz Coimbra de Pós-Graduação e Pesquisa de Engenharia

Instituto de Matemática

 
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Evento: 'Palestra do Professor Visitante Sándor Zoltán Németh'

Eventos PESC (Palestras, Seminários, etc.)
Palestras, Seminários, etc. do PESC/COPPE/UFRJ.
Data: Tuesday, August 19, 2014 At 13:00
Duração: 2 Horas

Sándor Zoltán Németh
School of Mathematics, University of Birmingham, UK

Período:18 a 21 de agosto de 2014

Palestra: 19 de agosto de 2014
Local: Sala H-316
Horário: 13h00

Título: 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 of simple Euclidean Jordan algebras of rank two is equivalent to determining the lattice-like sets with respect to a Lorentz cone.


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