Clarke Subgradients of Stratifiable Functions

Clarke Subgradients for Directionally Lipschitzian Stratifiable Functions

Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to the domain and limits of gradients generate the entire Clarke subdifferential. The characterization formula we obtain unifies various apparently disparate res...

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

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 ...

Method for solving generalized convex nonsmooth mixed-integer nonlinear programming problems

In this paper, we generalize the extended supporting hyperplane algorithm for a convex continuously differentiable mixed-integer nonlinear programming problem to solve a wider class of nonsmooth problems. The generalization is made by using the subgradients of the Clarke subdifferential instead of gradients. Consequently, all the functions in the problems are assumed to be locally Lipschitz con...

Approximating Functions on Stratifiable Sets

We investigate smooth approximations of functions, with prescribed gradient behavior on a distinguished stratifiable subset of the domain. As an application, we outline how our results yield important consequences for a recently introduced class of stochastic processes, called the matrix-valued Bessel processes.