Subtransversality and Strong CHIP of Closed Sets in Asplund Spaces

TL;DR

This paper investigates subtransversality and strong CHIP in Asplund spaces, providing characterizations, necessary conditions, and dual criteria for error bounds in optimization involving closed sets.

Contribution

It extends duality characterizations of subtransversality and strong CHIP from convex to non-convex sets in Asplund spaces, with applications to error bounds.

Findings

Characterizations of Asplund spaces via subtransversality.

Necessary conditions for subtransversality using normal cones.

Dual criteria for error bounds in inequality systems.

Abstract

In this paper, we mainly study subtransversality and two types of strong CHIP (given via Fr\'echet and limiting normal cones) for a collection of finitely many closed sets. We first prove characterizations of Asplund spaces in terms of subtransversality and intersection formulae of Fr\'echet normal cones. Several necessary conditions for subtransversality of closed sets are obtained via Fr\'echet/limiting normal cones in Asplund spaces. Then, we consider subtransversality for some special closed sets in convex-composite optimization. In this frame we prove an equivalence result on subtransversality, strong Fr\'echet CHIP and property (G) so as to extend a duality characterization of subtransversality of finitely many closed convex sets via strong CHIP and property (G) to the possibly non-convex case. As applications, we use these results on subtransversality and strong CHIP to study error bounds of inequality systems and give several dual criteria for error bounds via Fr\'echet normal cones and subdifferentials.