extremal principle – Optimization Online

Stationarity and regularity of infinite collections of sets

Published: 2011/12/19, Updated: 2012/11/12

Nonsmooth Optimization, Semi-infinite Programming asplund space, extremal principle, extremality, normal cone, optimality, regularity, stationarity, subdifferential

This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behaviour of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties … Read more

Local convergence for alternating and averaged nonconvex projections

Published: 2007/09/01

Adrian Lewis

Jérôme Malick

Russell Luke

Constrained Nonlinear Optimization, Control Applications, Nonsmooth Optimization alternating projection methods, averaged projections, distance to ill-posedness, extremal principle, linear convergence, metric regularity, nonconvex, prox-regularity, variational analysis

The idea of a finite collection of closed sets having “strongly regular intersection” at a given point is crucial in variational analysis. We show that this central theoretical tool also has striking algorithmic consequences. Specifically, we consider the case of two sets, one of which we assume to be suitably “regular” (special cases being convex … Read more