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The strong conical hull intersection property for convex programming. (English) Zbl 1134.90462

Summary: 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 1999, 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

$\left({P}_{f}\right)\phantom{\rule{1.em}{0ex}}min\left\{f\left(x\right)\mid x\in C,-g\left(x\right)\in S\right\}$

where $C$ is a closed convex subset of a Banach space $X$, $S$ is a closed convex cone which does not necessarily have non-empty interior, $Y$ is a Banach space, $f:X\to ℝ$ is a continuous convex function and $g:X\to Y$ is a continuous $S$-convex function. We also show that the strong CHIP completely characterizes the strong duality for partially finite convex programs, where $Y$ is finite dimensional and $g\left(x\right)=-Ax+b$ 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.

##### MSC:
 90C25 Convex programming 49J53 Set-valued and variational analysis
##### References:
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