Partial Smoothness, Subdifferentials and Set-valued Operators

TL;DR

This paper offers a new subdifferential-based characterization of partial smoothness, enhancing understanding and applicability in nonsmooth analysis and extending to set-valued operators.

Contribution

It introduces an alternative subdifferential-based characterization of partial smoothness, allowing for stronger identification results and generalization to set-valued operators.

Findings

Stronger identification results under degeneracy.

Explanation of identification with non-vanishing error.

Generalization to set-valued operators.

Abstract

Over the past decades, the concept "partial smoothness" has been playing as a powerful tool in several fields involving nonsmooth analysis, such as nonsmooth optimization, inverse problems and operation research, etc. The essence of partial smoothness is that it builds an elegant connection between the optimization variable and the objective function value through the subdifferential. Identifiability is the most appealing property of partial smoothness, as locally it allows us to conduct much finer or even sharp analysis, such as linear convergence or sensitivity analysis. However, currently the identifiability relies on non-degeneracy condition and exact dual convergence, which limits the potential application of partial smoothness. In this paper, we provide an alternative characterization of partial smoothness through only subdifferentials. This new perspective enables us to establish stronger identification results, explain identification under degeneracy and non-vanishing error. Moreover, we can generalize this new characterization to set-valued operators, and provide a complement definition of partly smooth operator proposed in [14].