328 research outputs found
Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees
We study the computational difficulty of the problem of finding fixed points
of nonexpansive mappings in uniformly convex Banach spaces. We show that the
fixed point sets of computable nonexpansive self-maps of a nonempty, computably
weakly closed, convex and bounded subset of a computable real Hilbert space are
precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A
uniform version of this result allows us to determine the Weihrauch degree of
the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is
equivalent to a closed choice principle, which receives as input a closed,
convex and bounded set via negative information in the weak topology and
outputs a point in the set, represented in the strong topology. While in finite
dimensional uniformly convex Banach spaces, computable nonexpansive mappings
always have computable fixed points, on the unit ball in infinite-dimensional
separable Hilbert space the Browder-Goehde-Kirk theorem becomes
Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is
equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive
mappings may not have any computable fixed points in infinite dimension. We
also study the computational difficulty of the problem of finding rates of
convergence for a large class of fixed point iterations, which generalise both
Halpern- and Mann-iterations, and prove that the problem of finding rates of
convergence already on the unit interval is equivalent to the limit operator.Comment: 44 page
Proof mining in metric fixed point theory and ergodic theory
In this survey we present some recent applications of proof mining to the
fixed point theory of (asymptotically) nonexpansive mappings and to the
metastability (in the sense of Terence Tao) of ergodic averages in uniformly
convex Banach spaces.Comment: appeared as OWP 2009-05, Oberwolfach Preprints; 71 page
An abstract proximal point algorithm
The proximal point algorithm is a widely used tool for solving a variety of
convex optimization problems such as finding zeros of maximally monotone
operators, fixed points of nonexpansive mappings, as well as minimizing convex
functions. The algorithm works by applying successively so-called "resolvent"
mappings associated to the original object that one aims to optimize. In this
paper we abstract from the corresponding resolvents employed in these problems
the natural notion of jointly firmly nonexpansive families of mappings. This
leads to a streamlined method of proving weak convergence of this class of
algorithms in the context of complete CAT(0) spaces (and hence also in Hilbert
spaces). In addition, we consider the notion of uniform firm nonexpansivity in
order to similarly provide a unified presentation of a case where the algorithm
converges strongly. Methods which stem from proof mining, an applied subfield
of logic, yield in this situation computable and low-complexity rates of
convergence
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