Dual Characterizations of Set Containments with Strict Convex Inequalities |
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Authors: | M A Goberna V Jeyakumar N Dinh |
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Institution: | (1) Department of Statistics and Operations Research, University of Alicante, 03080 Alicante, Spain;(2) Department of Applied Mathematics, University of New South Wales, Sydney, Australia;(3) Department of Mathematics-Informatics, Ho Chi Minh City University of Pedagogy, 280 An Duong Vuong Street, Distr. 5, HCM city, Vietman |
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Abstract: | Characterizations of the containment of a convex set either in an arbitrary convex set or in the complement of a finite union
of convex sets (i.e., the set, described by reverse-convex inequalities) are given. These characterizations provide ways of
verifying the containments either by comparing their corresponding dual cones or by checking the consistency of suitable associated
systems. The convex sets considered in this paper are the solution sets of an arbitrary number of convex inequalities, which
can be either weak or strict inequalities. Particular cases of dual characterizations of set containments have played key
roles in solving large scale knowledge-based data classification problems where they are used to describe the containments
as inequality constraints in optimization problems. The idea of evenly convex set (intersection of open half spaces), which
was introduced by W. Fenchel in 1952, is used to derive the dual conditions, characterizing the set containments. |
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Keywords: | Conjugacy Convex functions Dual cones Existence theorems Semi-infinite systems Set containment |
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