Identifying Activity

TL;DR

This paper explores how to identify active constraints in composite nonsmooth optimization problems, providing conditions for finite identification of active sets and extending classical constraint qualification concepts.

Contribution

It introduces a framework for activity identification in nonsmooth composite optimization, generalizing constraint qualification conditions to this broader setting.

Findings

Conditions for finite activity identification are established.

A generalized constraint qualification ensures boundedness of multipliers.

The approach extends classical optimization theory to nonsmooth, structured problems.

Abstract

Identification of active constraints in constrained optimization is of interest from both practical and theoretical viewpoints, as it holds the promise of reducing an inequality-constrained problem to an equality-constrained problem, in a neighborhood of a solution. We study this issue in the more general setting of composite nonsmooth minimization, in which the objective is a composition of a smooth vector function c with a lower semicontinuous function h, typically nonsmooth but structured. In this setting, the graph of the generalized gradient of h can often be decomposed into a union (nondisjoint) of simpler subsets. "Identification" amounts to deciding which subsets of the graph are "active" in the criticality conditions at a given solution. We give conditions under which any convergent sequence of approximate critical points finitely identifies the activity. Prominent among these properties is a condition akin to the Mangasarian-Fromovitz constraint qualification, which ensures boundedness of the set of multiplier vectors that satisfy the optimality conditions at the solution.