Restricted Normal Cones and the Method of Alternating Projections: Applications

Abstract

The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, Luke, and Malick derived local linear convergence results provided that a condition involving normal cones holds and at least one of the sets is superregular (a property less restrictive than convexity). However, their results failed to capture very simple classical convex instances such as two lines in three-dimensional space.In this paper, we extend and develop the Lewis-Luke-Malick framework so that not only any two linear subspaces but also any two closed convex sets whose relative interiors meet are covered. We also allow for sets that are more structured such as unions of convex sets.The key tool required is the restricted normal cone, which is a generalization of the classical Mordukhovich normal cone. We thoroughly study restricted normal cones from the viewpoint of constraint qualifications and regularity. Numerous examples are provided to illustrate the theory.2010 Mathematics Subject Classification: Primary 49J52, 49M20; Secondary 47H09, 65K05, 65K10, 90C26.

Transversality and Alternating Projections for Nonconvex Sets

“…It does not cover classically well-understood cases like two distinct intersecting lines in R 3 , a gap discussed in some recent work [1,4,5,17,34].…”

Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility

“…We show in (20) that one can easily calculate a projection onto . For closed and nonempty, we call the mapping the projector onto defined by (7) This is in general a set-valued mapping, indicated by the notation " " ( [38], Chapter 5). We call a point a projection.…”

Duality and Convex Programming

“…Note that the signal space is infinite dimensional while the measurement space is finite dimensional, a common situation in practice. Potter and Arun [100] recognized a much broader applicability of this variational formulation to remote sensing and medical imaging and applied duality theory to characterize solutions to (12) …”

“…Many of the algorithms for feasibility problems have counterparts for the more general best approximation problems [6,9,10,58,88]. For studies of these algorithms in nonconvex settings, see [2,7,8,[11][12][13][14]29,38,53,68,78,79,[87][88][89]. The projection algorithms that are central to convex feasibility and best approximation problems play a key role in algorithms for solving the problems considered here.…”