Relative Well-Posedness of Truncated Constrained Systems Accompanied by Variational Calculus

TL;DR

This paper develops new theoretical tools using variational analysis to characterize the well-posedness, stability, and regularity of truncated constrained systems in both finite and infinite-dimensional spaces.

Contribution

It introduces robust notions of relative contingent coderivatives and provides complete characterizations of well-posedness properties for constrained systems.

Findings

New characterizations of Lipschitz stability and metric regularity.

Introduction of relative contingent coderivatives.

Results applicable in both finite and infinite-dimensional settings.

Abstract

The paper concerns foundations of sensitivity and stability analysis in optimization and related areas, being primarily addressed truncated constrained systems. We consider general models, which are described by multifunctions between Banach spaces and concentrate on characterizing their well-posedness properties that revolve around Lipschitz stability and metric regularity relative to sets. Invoking tools of variational analysis and generalized differentiation, we introduce new robust notions of relative contingent coderivatives. The novel machinery of variational analysis leads us to establishing complete characterizations of such properties and developing basic rules of variational calculus interrelated with the obtained characterizations of well-posedness. Most of the our results valid in general infinite-dimensional settings are also new in finite dimensions.