Calculus of Tangent Sets and Derivatives of Set Valued Maps under Metric Subregularity Conditions

TL;DR

This paper develops calculus rules for tangent sets and derivatives of set-valued maps using metric subregularity, avoiding compactness assumptions, with applications to perturbation maps in optimization.

Contribution

It introduces a novel approach to calculus of tangent sets and derivatives based on metric subregularity, differing from existing methods by not requiring compactness.

Findings

Established calculus rules for tangent sets and derivatives under metric subregularity.

Extended the calculus to second order objects for set-valued maps.

Applied the results to perturbation maps in optimization problems.

Abstract

In this paper we intend to give some calculus rules for tangent sets in the sense of Bouligand and Ursescu, as well as for corresponding derivatives of set-valued maps. Both first and second order objects are envisaged and the assumptions we impose in order to get the calculus are in terms of metric subregularity of the assembly of the initial data. This approach is different from those used in alternative recent papers in literature and allows us to avoid compactness conditions. A special attention is paid for the case of perturbation set-valued maps which appear naturally in optimization problems.