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Strong CHIP for Infinite System of Closed Convex Sets in Normed Linear Spaces

Chong Li, K. F. Ng

2005
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SIAM Journal on Optimization
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For a general (possibly infinite) system of closed convex sets in a normed linear space we provide several sufficient conditions for ensuring the strong conical hull intersection property. One set of sufficient conditions is given in terms of the finite subsystems while the other sets are in terms of the relaxed interior-point conditions together with appropriate continuity of the associated setvalued function on the (topologized) index set I. In the special case when I is finite and X is
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... nite and X is finite dimensional, one of these results reduces to a classical result of Rockafellar. Introduction. The notion of the strong CHIP (conical hull intersection property) was introduced by Deutsch, Li, and Ward in [12, 13] for a finite family of closed convex sets in a Euclidean space (or a Hilbert space) and has been successfully applied in the reformulation of some best approximation problems. This notion closely relates other fundamental concepts such as bounded linear regularity, G-property of Jameson, error bounds in convex optimization [1, 3] , and the BCQ (basic constraint qualification) as well as the perturbations for finite convex systems of inequalities. See [5, 6, 8, 12, 13, 14, 18, 19, 24, 25] and references therein, especially in [20] , where the strong CHIP was defined for an arbitrary family of closed convex sets in a Banach space and utilized in the study of general systems of infinite convex inequalities, such as the system that naturally arises from the problem of best restricted range approximation in the space C(Q) of complex-valued continuous functions on a compact metric space Q under quite general constraints. This problem was first presented and formulated by Smirnov and Smirnov in [31, 32] , where each Ω t was assumed to be a disk in C. Later in [33, 34, 35] and also more recently in [17, 20] , the constraint sets Ω t have been relaxed but still remain to assume the strong interior-point condition (in particular, int Ω t = ∅ for each t ∈ Q). This unfortunately excludes the interesting case when some Ω t is a line segment or a singleton in C. As demonstrated in an accompanying paper [22] , the results obtained in the present paper have enabled us to study the restricted range approximation problem under much less restrictive assumptions by allowing the case that int Ω t = ∅ for some t ∈ Q. The present paper is devoted to providing sufficient conditions for a (finite or infinite) family {C, C i : i ∈ I} of closed convex sets in a Banach (or normed linear) space to have the strong CHIP. In expanding and improving the known results on the sufficient conditions for the * , z ≤ 0 for all z ∈ Z}. The normal cone of Z at z 0 is denoted by N Z (z 0 ) and defined by N Z (z 0 ) = (Z − z 0 ) . For convenience of printing we sometimes use N (z 0 ; Z) in place of N Z (z 0 ). Let A be a closed convex nonempty subset of X. The interior and boundary of Z relative to A are denoted by rint A Z and bd A Z, respectively; they are defined to be, respectively, the interior and boundary of the set aff A ∩ Z in the metric space aff A. Thus, a point z ∈ rint A Z if and only if there exists ε > 0 such that z ∈ (aff A) ∩ B(z, ε) ⊆ Z, (2.1) while z ∈ bd A Z if and only if z ∈ aff A and, for any ε > 0, (aff A) ∩ B(z, ε) intersects Z and its complement. Let R − denote the subset of R consisting of all nonpositive real

doi:10.1137/040613238
fatcat:xml74ddiibchtcvbbfscqdk6aq