The Clarke Generalized Gradient for Functions Whose Epigraph Has Positive Reach
Giovanni Colombo, Antonio Marigonda, Peter R. Wolenski · Mathematics of Operations Research · 2013
We consider the class of continuous functions that map an open set Ω ⫅ ℝ n to ℝ with an epigraph having (locally) positive reach with an additional property. This class contains all finite convex and C 1, 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.