Regularity conditions for
the linear separation of sets

Giancarlo Bigi and Massimo Pappalardo


In recent papers, a concept of regularity for linear separation between a set K and a convex cone H has been introduced and characterized. Regular separation does not prevent the existence of an irregular one; thus, a concept of total regularity, ensuring that only regular separation holds, has been here investigated. We point out that regularity and total regularity conditions strengthen the concept of proper separation. In the same papers, constrained extremum problems have been analysed within this framework through generalized systems and image space approach; this kind of analysis has led to constraint qualifications and regularity conditions for Karush-Kuhn-Tucker multipliers, which generalize well known ones. In this paper we extend the analysis of linear separation to a more general setting. We want to deepen the study of proper separation, concentrating our attention to the inclusion of a generic face of the cone H into separating hyperplanes. In particular, we are interested in conditions which ensure that the given face is not included in at least one or in every separating hyperplane. To this aim we introduce the concepts of regularity and total regularity with respect to a face and we characterize them.

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