Directional quasi/pseudo-normality as sufficient conditions for metric subregularity

TL;DR

This paper introduces new directional normality conditions that serve as sufficient criteria for metric subregularity in set-valued maps, extending existing concepts and applicable to optimization systems.

Contribution

It proposes a directional pseudo-normality condition, weaker than classical conditions, and demonstrates its effectiveness and automatic validity in certain structured problems.

Findings

Directional pseudo-normality can be weaker than FOSCMS.

The new conditions are automatically satisfied for affine maps with polyhedral sets.

Applications include complementarity and KKT systems.

Abstract

In this paper we study sufficient conditions for metric subregularity of a set-valued map which is the sum of a single-valued continuous map and a locally closed subset. First we derive a sufficient condition for metric subregularity which is weaker than the so-called first-order sufficient condition for metric subregularity (FOSCMS) by adding an extra sequential condition. Then we introduce a directional version of the quasi-normality and the pseudo-normality which is stronger than the new {weak} sufficient condition for metric subregularity but is weaker than the classical quasi-normality and pseudo-normality respectively. Moreover we introduce a nonsmooth version of the second-order sufficient condition for metric subregularity and show that it is a sufficient condition for the new sufficient condition for metric {sub}regularity to hold. An example is used to illustrate that the directional pseduo-normality can be weaker than FOSCMS. For the class of set-valued maps where the single-valued mapping is affine and the abstract set is the union of finitely many convex polyhedral sets, we show that the pseudo-normality and hence the directional pseudo-normality holds automatically at each point of the graph. Finally we apply our results to the complementarity and the Karush-Kuhn-Tucker systems.