29 research outputs found
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
A Note on the Finite Convergence of Alternating Projections
We establish sufficient conditions for finite convergence of the alternating
projections method for two non-intersecting and potentially nonconvex sets. Our
results are based on a generalization of the concept of intrinsic
transversality, which until now has been restricted to sets with nonempty
intersection. In the special case of a polyhedron and closed half space, our
sufficient conditions define the minimum distance between the two sets that is
required for alternating projections to converge in a single iteration.Comment: 9 pages, 7 figure
Error Bounds and Holder Metric Subregularity
The Holder setting of the metric subregularity property of set-valued
mappings between general metric or Banach/Asplund spaces is investigated in the
framework of the theory of error bounds for extended real-valued functions of
two variables. A classification scheme for the general Holder metric
subregularity criteria is presented. The criteria are formulated in terms of
several kinds of primal and subdifferential slopes.Comment: 32 pages. arXiv admin note: substantial text overlap with
arXiv:1405.113