  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.
