The U-Lagrangian of a prox-regular function

TL;DR

This paper extends the concept of U-Lagrangian and UV-decomposition from convex to nonconvex prox-regular functions, enabling smoother analysis of nonsmooth functions in subspaces.

Contribution

It generalizes the U-Lagrangian and UV-decomposition framework to nonconvex prox-regular functions, broadening their applicability.

Findings

Defines U-Lagrangian for prox-regular functions.

Establishes properties of the generalized UV-decomposition.

Provides theoretical insights into nonsmooth nonconvex optimization.

Abstract

When restricted to a subspace, a nonsmooth function can be differentiable. It is known that for a nonsmooth convex function f and a point x, the Euclidean space can be decomposed into two subspaces: U, over which a special Lagrangian can be defined and has nice smooth properties and V, the orthogonal complement subspace of U. In this paper we generalize the definition of UV-decomposition and U-Lagrangian to the context of nonconvex functions, specifically that of a prox-regular function.