The strong conical hull intersection property for convex programming
 V. Jeyakumar
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The strong conical hull intersection property (CHIP) is a geometric property of a collection of finitely many closed convex intersecting sets. This basic property, which was introduced by Deutsch et al. in 1997, is one of the central ingredients in the study of constrained interpolation and best approximation. In this paper we establish that the strong CHIP of intersecting sets of constraints is the key characterizing property for optimality and strong duality of convex programming problems. We first show that a sharpened strong CHIP is necessary and sufficient for a complete Lagrange multiplier characterization of optimality for the convex programming model problem
where C is a closed convex subset of a Banach space X, S is a closed convex cone which does not necessarily have nonempty interior, Y is a Banach space, is a continuous convex function and g:X→Y is a continuous Sconvex function. We also show that the strong CHIP completely characterizes the strong duality for partially finite convex programs, where Y is finite dimensional and g(x)=−Ax+b and S is a polyhedral convex cone. Global sufficient conditions which are strictly weaker than the Slater type conditions are given for the strong CHIP and for the sharpened strong CHIP.
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 Title
 The strong conical hull intersection property for convex programming
 Journal

Mathematical Programming
Volume 106, Issue 1 , pp 8192
 Cover Date
 20060301
 DOI
 10.1007/s1010700506054
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Strong conical hull intersection property
 global constraint qualification
 strong duality
 optimality conditions
 constrained approximation
 41A65
 41A29
 90C30
 Industry Sectors
 Authors

 V. Jeyakumar ^{(1)}
 Author Affiliations

 1. Department of Applied Mathematics, University of New South Wales, Sydney, 2052, Australia