The strong conical hull intersection property for convex programming

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 [inline-graphic not available: see fulltext]