Generalized Differentiation for Moving Objects

Abstract

This work is devoted to the study of the analysis of moving sets/mappings in the form \(\{\Omega_ t \}_{ t \in T}\) or \(\{F_ t \}_{ t \in T}\). Such objects arise commonly both in applications and theories including parameterized optimization, non-autonomous control, mathematical economics, etc. As the principal part of nonsmooth variational analysis, the generalized diSerentiation theory for the non-moving objects has been developed greatly in the past few decades, which greatly advanced the related applications and studies. Being a natural extension and much required by applications, the study of the corresponding moving objects, however, has been quite limited. In this paper, we summarize and generalize some of the recently developed concepts, namely, extended normal cone, extended coderivative and extended subdifferential, which are natural generalizations of the corresponding normal cone, coderivative and subdifferential for non-moving objects, respectively . We then develop complete calculus rules for such extended constructions. Similarly to the case of non-moving objects, the developed calculus of extended constructions requires certain normal compactness conditions in infinite-dimensional spaces. As the second goal of the paper, we propose new sequential normal compactness for the moving objects and then establish calculus rules for the extended normal compactness that are crucial to applications. We illustrate that most results for the non-moving situations could be generalized to the moving case. The main results are developed in Asplund spaces, and our main tool is a fuzzy intersection rule based on the extremal principle.