58 research outputs found
Order preserving and order reversing operators on the class of convex functions in Banach spaces
A remarkable result by S. Artstein-Avidan and V. Milman states that, up to
pre-composition with affine operators, addition of affine functionals, and
multiplication by positive scalars, the only fully order preserving mapping
acting on the class of lower semicontinuous proper convex functions defined on
is the identity operator, and the only fully order reversing one
acting on the same set is the Fenchel conjugation. Here fully order preserving
(reversing) mappings are understood to be those which preserve (reverse) the
pointwise order among convex functions, are invertible, and such that their
inverses also preserve (reverse) such order. In this paper we establish a
suitable extension of these results to order preserving and order reversing
operators acting on the class of lower semicontinous proper convex functions
defined on arbitrary infinite dimensional Banach spaces.Comment: 19 pages; Journal of Functional Analysis, accepted for publication; a
better presentation of certain parts; minor corrections and modifications;
references and thanks were adde
On the centralization of the circumcentered-reflection method
This paper is devoted to deriving the first circumcenter iteration scheme
that does not employ a product space reformulation for finding a point in the
intersection of two closed convex sets. We introduce a so-called centralized
version of the circumcentered-reflection method (CRM). Developed with the aim
of accelerating classical projection algorithms, CRM is successful for tracking
a common point of a finite number of affine sets. In the case of general convex
sets, CRM was shown to possibly diverge if Pierra's product space reformulation
is not used. In this work, we prove that there exists an easily reachable
region consisting of what we refer to as centralized points, where pure
circumcenter steps possess properties yielding convergence. The resulting
algorithm is called centralized CRM (cCRM). In addition to having global
convergence, cCRM converges linearly under an error bound condition, and
superlinearly if the two target sets are so that their intersection have
nonempty interior and their boundaries are locally differentiable manifolds. We
also run numerical experiments with successful results.Comment: 29 pages with 7 figure
A successive centralized circumcenter reflection method for the convex feasibility problem
In this paper we present the successive centralization of the circumcenter
reflection scheme with several control sequences for solving the convex
feasibility problem in Euclidean space. Assuming that a standard error bound
holds, we prove the linear convergence of the method with the most violated
constraint control sequence. Under additional smoothness assumptions, we prove
the superlinear convergence. Numerical experiments confirm the efficiency of
our method
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