344 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
Local Linear Convergence of Approximate Projections onto Regularized Sets
The numerical properties of algorithms for finding the intersection of sets
depend to some extent on the regularity of the sets, but even more importantly
on the regularity of the intersection. The alternating projection algorithm of
von Neumann has been shown to converge locally at a linear rate dependent on
the regularity modulus of the intersection. In many applications, however, the
sets in question come from inexact measurements that are matched to idealized
models. It is unlikely that any such problems in applications will enjoy
metrically regular intersection, let alone set intersection. We explore a
regularization strategy that generates an intersection with the desired
regularity properties. The regularization, however, can lead to a significant
increase in computational complexity. In a further refinement, we investigate
and prove linear convergence of an approximate alternating projection
algorithm. The analysis provides a regularization strategy that fits naturally
with many ill-posed inverse problems, and a mathematically sound stopping
criterion for extrapolated, approximate algorithms. The theory is demonstrated
on the phase retrieval problem with experimental data. The conventional early
termination applied in practice to unregularized, consistent problems in
diffraction imaging can be justified fully in the framework of this analysis
providing, for the first time, proof of convergence of alternating approximate
projections for finite dimensional, consistent phase retrieval problems.Comment: 23 pages, 5 figure
A convergent relaxation of the Douglas-Rachford algorithm
This paper proposes an algorithm for solving structured optimization
problems, which covers both the backward-backward and the Douglas-Rachford
algorithms as special cases, and analyzes its convergence. The set of fixed
points of the algorithm is characterized in several cases. Convergence criteria
of the algorithm in terms of general fixed point operators are established.
When applying to nonconvex feasibility including the inconsistent case, we
prove local linear convergence results under mild assumptions on regularity of
individual sets and of the collection of sets which need not intersect. In this
special case, we refine known linear convergence criteria for the
Douglas-Rachford algorithm (DR). As a consequence, for feasibility with one of
the sets being affine, we establish criteria for linear and sublinear
convergence of convex combinations of the alternating projection and the DR
methods. These results seem to be new. We also demonstrate the seemingly
improved numerical performance of this algorithm compared to the RAAR algorithm
for both consistent and inconsistent sparse feasibility problems
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