29,080 research outputs found
Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods
The convex feasibility problem (CFP) is at the core of the modeling of many
problems in various areas of science. Subgradient projection methods are
important tools for solving the CFP because they enable the use of subgradient
calculations instead of orthogonal projections onto the individual sets of the
problem. Working in a real Hilbert space, we show that the sequential
subgradient projection method is perturbation resilient. By this we mean that
under appropriate conditions the sequence generated by the method converges
weakly, and sometimes also strongly, to a point in the intersection of the
given subsets of the feasibility problem, despite certain perturbations which
are allowed in each iterative step. Unlike previous works on solving the convex
feasibility problem, the involved functions, which induce the feasibility
problem's subsets, need not be convex. Instead, we allow them to belong to a
wider and richer class of functions satisfying a weaker condition, that we call
"zero-convexity". This class, which is introduced and discussed here, holds a
promise to solve optimization problems in various areas, especially in
non-smooth and non-convex optimization. The relevance of this study to
approximate minimization and to the recent superiorization methodology for
constrained optimization is explained.Comment: Mathematical Programming Series A, accepted for publicatio
On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets
The Krein-Milman theorem (1940) states that every convex compact subset of a
Hausdorfflocally convex topological space, is the closed convex hull of its
extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the
general framework of -convexity. Under general conditions on the class of
functions , the Krein-Milman-Ky Fan theorem asserts then, that every
compact -convex subset of a Hausdorff space, is the -convex hull of
its -extremal points. We prove in this paper that, in the metrizable case
the situation is rather better. Indeed, we can replace the set of
-extremal points by the smaller subset of -exposed points. We
establish under general conditions on the class of functions , that every
-convex compact metrizable subset of a Hausdorff space, is the
-convex hull of its -exposed points. As a consequence we obtain
that each convex weak compact metrizable (resp. convex weak compact
metrizable) subset of a Banach space (resp. of a dual Banach space), is the
closed convex hull of its exposed points (resp. the weak closed convex hull
of its weak exposed points). This result fails in general for compact
-convex subsets that are not metrizable
Two Forms of Inconsistency in Quantum Foundations
Recently, there has been some discussion of how Dutch Book arguments might be
used to demonstrate the rational incoherence of certain hidden variable models
of quantum theory (Feintzeig and Fletcher 2017). In this paper, we argue that
the 'form of inconsistency' underlying this alleged irrationality is deeply and
comprehensively related to the more familiar 'inconsistency' phenomenon of
contextuality. Our main result is that the hierarchy of contextuality due to
Abramsky and Brandenburger (2011) corresponds to a hierarchy of
additivity/convexity-violations which yields formal Dutch Books of different
strengths. We then use this result to provide a partial assessment of whether
these formal Dutch Books can be interpreted normatively.Comment: 26 pages, 5 figure
Existence of Minimizers for Non-Level Convex Supremal Functionals
The paper is devoted to determine necessary and sufficient conditions for
existence of solutions to the problem when the supremand is
not necessarily level convex. These conditions are obtained through a
comparison with the related level convex problem and are written in terms of a
differential inclusion involving the boundary datum. Several conditions of
convexity for the supremand are also investigated
- …