4,515 research outputs found
Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems
We consider projection algorithms for solving (nonconvex) feasibility
problems in Euclidean spaces. Of special interest are the Method of Alternating
Projections (MAP) and the Douglas-Rachford or Averaged Alternating Reflection
Algorithm (AAR). In the case of convex feasibility, firm nonexpansiveness of
projection mappings is a global property that yields global convergence of MAP
and for consistent problems AAR. Based on (\epsilon, \delta)-regularity of sets
developed by Bauschke, Luke, Phan and Wang in 2012, a relaxed local version of
firm nonexpansiveness with respect to the intersection is introduced for
consistent feasibility problems. Together with a coercivity condition that
relates to the regularity of the intersection, this yields local linear
convergence of MAP for a wide class of nonconvex problems,Comment: 22 pages, no figures, 30 reference
Approximation of *weak-to-norm continuous mappings
The purpose of this paper is to study the approximation of vector valued
mappings defined on a subset of a normed space. We investigate Korovkin-type
conditions under which a given sequence of linear operators becomes a so-called
approximation process. First, we give a sufficient condition for this sequence
to approximate the class of bounded, uniformly continuous functions. Then we
present some sufficient and necessary conditions guaranteeing the approximation
within the class of unbounded, *weak-to-norm continuous mappings. We also
derive some estimates of the rate of convergence.Comment: 13 page
- …