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Identifying Activity

Adrian S. Lewis (aslewis***at***orie.cornell.edu)
Stephen J. Wright (swright***at***cs.wisc.edu)

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.

Keywords: constrained optimization, composite optimization, Mangasarian-Fromovitz constraint qualification, active set, identification

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Citation: ORIE Technical Report, Cornell University, January 2009. Revised December 2010.

Download: [PDF]

Entry Submitted: 01/17/2009
Entry Accepted: 01/17/2009
Entry Last Modified: 01/13/2011

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