The Clarke Generalized Gradient for Functions Whose Epigraph Has Positive Reach

• 成果类型: Article
• 署名作者: Colombo, Giovanni; Marigonda, Antonio; Wolenski, Peter R.
• 署名单位: University of Padua; University of Verona; Louisiana State University System; Louisiana State University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.1120.0580
• 发表日期: 2013
• 页码: 451-468
• 关键词: regularity properties; sets; differentiability

摘要

We consider the class of continuous functions that map an open set Omega subset of R-n to R with an epigraph having (locally) positive reach with an additional property. This class contains all finite convex and C-1,C- 1 functions, but also ones that are not necessarily Lipschitz continuous. We provide a representation formula for the Clarke generalized gradient of such functions using convex combinations and limits of gradients at differentiability points, thus offering an alternative to the well-known proximal normal formula by replacing a pointedness assumption by one of positive reach. Our proof consists of a detailed analysis of singularities using methods taken from both nonsmooth analysis and geometric measure theory, and is based on an induction argument. As an application, we prove for a particular class of Hamilton-Jacobi equations that an a.e. solution whose hypograph has positive reach and satisfies an additional property is indeed the unique viscosity solution.