Restricted Normal Cones and the Method of Alternating Projections: Applications

By Heinz H. Bauschke, D. R. Luke, Hung M. Phan, Xianfu Wang. 2012

61 citations recorded by the source corpus.

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 a 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. Numerous examples are provided to illustrate the theory.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/070d4e6388e0b84bc78204af463fadcfe504c70a

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In Applied Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Linear convergence of the Douglas–Rachford method for two closed sets (2014, unknown): “ containing A, then the (limiting) normal cone of A restricted to L is given by (9) NL A(x) := NA(x) ∩(L − x). Clearly, NL A(x) ⊆ NA(x). Indeed, the concept of restricted normal cone was developed in [8, 9, 11] in more general settings. Deﬁnition 2.1 (regularity of set systems) Let A and B be two su”
Restricted Normal Cones and the Method of Alternating Projections: Theory (2012, methodology): “4, it is beneficial to work with a suitable partition to obtain a CQ-number that is less than one, which in turn is very desirable in applications (see [3] and [4]).”