Partial smoothness and constant rank

TL;DR

This paper clarifies that partial smoothness in optimization, which helps algorithms identify active constraints, is essentially a constant-rank condition in differential geometry, extending to broader mappings like saddlepoint operators.

Contribution

It simplifies the concept of partial smoothness to a constant-rank condition, broadening its applicability to various mappings in optimization.

Findings

Partial smoothness is equivalent to a constant-rank condition.

This perspective simplifies the analysis of active set identification.

Extension to saddlepoint operators in primal-dual algorithms.

Abstract

The idea of partial smoothness in optimization blends certain smooth and nonsmooth properties of feasible regions and objective functions. As a consequence, the standard first-order conditions guarantee that diverse iterative algorithms (and post-optimality analyses) identify active structure or constraints. However, by instead focusing directly on the first-order conditions, the formal concept of partial smoothness simplifies dramatically: in basic differential geometric language, it is just a constant-rank condition. In this view, partial smoothness extends to more general mappings, such as saddlepoint operators underlying primal-dual splitting algorithms.