On a strong covering property of multivalued mappings

TL;DR

This paper introduces a strong covering property for multivalued mappings, explores its stability, and applies it to set-valued inclusions and optimization problems, providing new insights into their solvability and perturbation behavior.

Contribution

It defines a new strong covering property for multivalued mappings, characterizes a class of convex processes satisfying this property, and studies its stability and applications.

Findings

Identifies a class of closed convex processes with the covering property

Shows the property is stable under Lipschitz perturbations

Applies the property to solvability of set-valued inclusions and optimization

Abstract

In this paper, a strong variant for multivalued mappings of the well-known property of openness at a linear rate is studied. Among other examples, a simply characterized class of closed convex processes between Banach spaces, which satisfies such a covering behaviour, is singled out. Equivalent reformulations of this property and its stability under Lipschitz perturbations are investigated in a metric space setting. Applications to the solvability of set-valued inclusions and to the exact penalization of optimization problems with set-inclusion constraints are discussed.