The notion of a strictly maximal point is a concept of proper maximality that plays an important role in the study of the stability of vector optimization problems. The aim of this paper is to study some properties of this notion with particular attention to geometrical aspects. More precisely, we individuate some relationships between strict maximality and the properties of the bases of the ordering cone. In order to prove this result, a new characterization of the existence of a bounded base for a closed convex cone is given. Moreover, we link strict maximality to the geometrical notion of strongly exposed points of a given set. Finally, we deal with the linear scalarization for the strictly maximal points.
Casini, E., Miglierina, E., The Geometry of Strict Maximality, <<SIAM JOURNAL ON OPTIMIZATION>>, 2010; 20 (6): 3146-3160. [doi:10.1137/090748858] [http://dx.medra.org/10.1137/090748858] [http://hdl.handle.net/10807/1426]
The Geometry of Strict Maximality
Miglierina, Enrico
2010
Abstract
The notion of a strictly maximal point is a concept of proper maximality that plays an important role in the study of the stability of vector optimization problems. The aim of this paper is to study some properties of this notion with particular attention to geometrical aspects. More precisely, we individuate some relationships between strict maximality and the properties of the bases of the ordering cone. In order to prove this result, a new characterization of the existence of a bounded base for a closed convex cone is given. Moreover, we link strict maximality to the geometrical notion of strongly exposed points of a given set. Finally, we deal with the linear scalarization for the strictly maximal points.
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