Bounded Linear Regularity, Strong CHIP, and CHIP are Distinct Properties
نویسندگان
چکیده
Bounded linear regularity, the strong conical hull intersection property (strong CHIP), and the conical hull intersection property (CHIP) are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. It was shown recently that these properties are fundamental in several branches of convex optimization, including convex feasibility problems, error bounds, Fenchel duality, and constrained approximation. It was known that regularity implies strong CHIP, which in turn implies CHIP; moreover, the three properties always hold for subspaces. The question whether or not converse implications are true for general convex sets was open.
منابع مشابه
Strong Chip, Normality, and Linear Regularity of Convex Sets
We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a nite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex cones, the dual normal property is the proper...
متن کاملStrong Topological Regularity and Weak Regularity of Banach Algebras
In this article we study two different generalizations of von Neumann regularity, namely strong topological regularity and weak regularity, in the Banach algebra context. We show that both are hereditary properties and under certain assumptions, weak regularity implies strong topological regularity. Then we consider strong topological regularity of certain concrete algebras. Moreover we obtain ...
متن کاملThe SECQ Linear Regularity and the Strong CHIP for In nite System of Closed Convex Sets in Normed Linear Spaces
We consider a nite or in nite family of closed convex sets with nonempty intersection in a normed space A property relating their epigraphs with their intersection s epigraph is studied and its relations to other constraint quali cations such as the linear regularity the strong CHIP and Jameson s G property are estab lished With suitable continuity assumption we show how this property can be en...
متن کاملRegularity of Bounded Tri-Linear Maps and the Fourth Adjoint of a Tri-Derivation
In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if f : X × Y × Z −→ W is a bounded tri-linear mapping and h : W −→ S is a bounded linear mapping, then f is regular if and only if hof is regular. We also shall give some necessary and sufficient conditions such that the fourth adjoint D^∗∗∗∗ of a tri-derivation D is again tri-derivation.
متن کاملThe SECQ, Linear Regularity, and the Strong CHIP for an Infinite System of Closed Convex Sets in Normed Linear Spaces
We consider a (finite or infinite) family of closed convex sets with nonempty intersection in a normed space. A property relating their epigraphs with their intersection’s epigraph is studied, and its relations to other constraint qualifications (such as the linear regularity, the strong CHIP, and Jameson’s (G)-property) are established. With suitable continuity assumption we show how this prop...
متن کامل