5,542 research outputs found
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
Transversality Properties: Primal Sufficient Conditions
The paper studies 'good arrangements' (transversality properties) of
collections of sets in a normed vector space near a given point in their
intersection. We target primal (metric and slope) characterizations of
transversality properties in the nonlinear setting. The Holder case is given a
special attention. Our main objective is not formally extending our earlier
results from the Holder to a more general nonlinear setting, but rather to
develop a general framework for quantitative analysis of transversality
properties. The nonlinearity is just a simple setting, which allows us to unify
the existing results on the topic. Unlike the well-studied subtransversality
property, not many characterizations of the other two important properties:
semitransversality and transversality have been known even in the linear case.
Quantitative relations between nonlinear transversality properties and the
corresponding regularity properties of set-valued mappings as well as nonlinear
extensions of the new transversality properties of a set-valued mapping to a
set in the range space due to Ioffe are also discussed.Comment: 33 page
About [q]-regularity properties of collections of sets
We examine three primal space local Hoelder type regularity properties of
finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and
uniform [q]-regularity as well as their quantitative characterizations.
Equivalent metric characterizations of the three mentioned regularity
properties as well as a sufficient condition of [q]-subregularity in terms of
Frechet normals are established. The relationships between [q]-regularity
properties of collections of sets and the corresponding regularity properties
of set-valued mappings are discussed.Comment: arXiv admin note: substantial text overlap with arXiv:1309.700
An induction theorem and nonlinear regularity models
A general nonlinear regularity model for a set-valued mapping , where and are metric spaces, is considered
using special iteration procedures, going back to Banach, Schauder, Lusternik
and Graves. Namely, we revise the induction theorem from Khanh, J. Math. Anal.
Appl., 118 (1986) and employ it to obtain basic estimates for studying
regularity/openness properties. We also show that it can serve as a
substitution of the Ekeland variational principle when establishing other
regularity criteria. Then, we apply the induction theorem and the mentioned
estimates to establish criteria for both global and local versions of
regularity/openness properties for our model and demonstrate how the
definitions and criteria translate into the conventional setting of a
set-valued mapping .Comment: 28 page
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