Regularities and their relations to error bounds
In this paper. we mainly study various notions of regularity for a finite collection (C-1,...,C-m) of closed convex Subsets of a Barrach space X and their relations with other fundamental concepts. We show that a proper lower semicontinuous function f on X has a Lipschitz error bound (resp., Y-error bound) if and only if the pair {epi (f), X x {0}} of sets in the product space X x R is linearly regular (resp., regular). Similar results for multifunctions are also established. Next. we prove that {C-1,...,C-m} is linearly regular if and onIy if it has the strong CHIP and the collection {N-C1 (z),...,N-Cm (z)} of normal cones at z has property (G) for each z is an element of C := boolean AND(i=1)(m) C-i. Provided that C-1 is a closed convex cone and that C-2 = Y is a closed vector subspace of X, we show that (C-1, Y) is linearly regular if and only if there exists alpha > 0 such that each positive (relative to the order induced by C,) linear functional on Y of norm one can be extended to a positive linear functional on X with norm bounded by alpha. Similar characterization is given in terms of normal cones.
regular error bound linearly regular multifunction strong chip normal cone property (g) positive linear functional epigraph graph tangent cone