Identifying Activity
Adrian S. Lewis, Stephen J. Wright · SIAM Journal on Optimization · 2011
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 [Formula: see text] with a lower semicontinuous function [Formula: see text], typically nonsmooth but structured. In this setting, the graph of the generalized gradient [Formula: see text] can often be decomposed into a (nondisjoint) union 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.