Clarke Subgradients for Directionally Lipschitzian Stratifiable Functions

Clarke Subgradients of Stratifiable Functions

We establish the following result: if the graph of a lower semicontinuous real-extendedvalued function f : Rn → R ∪ {+∞} admits a Whitney stratification (so in particular if f is a semialgebraic function), then the norm of the gradient of f at x ∈ dom f relative to the stratum containing x bounds from below all norms of Clarke subgradients of f at x. As a consequence, we obtain a Morse-Sard typ...

An effective optimization algorithm for locally nonconvex Lipschitz functions based on mollifier subgradients

Various Lipschitz-like Properties for Functions and Sets I: Directional Derivative and Tangential Characterizations

In this work we introduce for extended real valued functions, defined on a Banach space X, the concept of K directionally Lipschitzian behavior, where K is a bounded subset of X. For different types of sets K (e.g., zero, singleton, or compact), the K directionally Lipschitzian behavior recovers well-known concepts in variational analysis (locally Lipschitzian, directionally Lipschitzian, or co...

Analysis of Direct Searches for Non-Lipschitzian Functions

It is known that the Clarke generalized directional derivative is nonnegative along the limit directions generated by directional directsearch methods at a limit point of certain subsequences of unsuccessful iterates, if the function being minimized is Lipschitz continuous near the limit point. In this paper we generalize this result for non-Lipschitzian functions using Rockafellar generalized ...

An Effective Optimization Algorithm for Locally Nonconvex Lipschitz Functions Based on Mollifier Subgradients

We present an effective algorithm for minimization of locally nonconvex Lipschitz functions based on mollifier functions approximating the Clarke generalized gradient. To this aim, first we approximate the Clarke generalized gradient by mollifier subgradients. To construct this approximation, we use a set of averaged functions gradients. Then, we show that the convex hull of this set serves as ...