Generalized differentiation of parameter-dependent sets and mappings†

†Dedicated to Bert Jongen in honor of his 60th birthday.

Abstract

This article concerns new aspects of generalized differentiation theory that plays a crucial role in many areas of modern variational analysis, optimization and their applications. In contrast to the majority of previous developments, we focus here on generalized differentiation of parameter-dependent objects (sets, set-valued mappings and nonsmooth functions), which naturally appear, e.g. in parametric optimization and related topics. The basic generalized differential constructions needed in this case are different for those known in parameter-independent settings, while they still enjoy comprehensive calculus rules developed in this article.

†Dedicated to Bert Jongen in honor of his 60th birthday.

Keywords:

• nonsmooth and parametric optimization
• variational analysis
• generalized differentiation
• calculus rules
• normal compactness

Mathematics Subject Classifications 2000: :

• 49J52
• 49J53; Secondary: 90C29

Acknowledgements

Research was partially supported by the National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP-0451168.

Notes

†Dedicated to Bert Jongen in honor of his 60th birthday.