Universidade Federal do Rio de Janeiro COPPE
Instituto Alberto Luiz Coimbra de PósGraduação e Pesquisa de Engenharia
Instituto de Matemática




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 H316 Horário: 13h00 Título: Latticelike 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 selfdual cone and in particular onto the cone of squares. We have extended these operations to more general cones. We have shown that these latticelike 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 latticelike operations, called latticelike 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 latticelike 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 latticelike with respect to a simplicial cone. We have considered the problem of latticelike sets in Euclidean Jordan algebras with respect to the cone of squares. We have shown that the Jordan subalgebras are latticelike sets, but the converse in general is not true. In the case of simple Euclidean Jordan algebras of rank at least three the latticelike property is rather restrictive, e.g., there are no latticelike proper closed convex sets with interior points. The case of of simple Euclidean Jordan algebras of rank two is equivalent to determining the latticelike sets with respect to a Lorentz cone. 


