3 research outputs found
The Radius of Metric Subregularity
There is a basic paradigm, called here the radius of well-posedness, which
quantifies the "distance" from a given well-posed problem to the set of
ill-posed problems of the same kind. In variational analysis, well-posedness is
often understood as a regularity property, which is usually employed to measure
the effect of perturbations and approximations of a problem on its solutions.
In this paper we focus on evaluating the radius of the property of metric
subregularity which, in contrast to its siblings, metric regularity, strong
regularity and strong subregularity, exhibits a more complicated behavior under
various perturbations. We consider three kinds of perturbations: by Lipschitz
continuous functions, by semismooth functions, and by smooth functions,
obtaining different expressions/bounds for the radius of subregularity, which
involve generalized derivatives of set-valued mappings. We also obtain
different expressions when using either Frobenius or Euclidean norm to measure
the radius. As an application, we evaluate the radius of subregularity of a
general constraint system. Examples illustrate the theoretical findings.Comment: 20 page
H\"older Error Bounds and H\"older Calmness with Applications to Convex Semi-Infinite Optimization
Using techniques of variational analysis, necessary and sufficient
subdifferential conditions for H\"older error bounds are investigated and some
new estimates for the corresponding modulus are obtained. As an application, we
consider the setting of convex semi-infinite optimization and give a
characterization of the H\"older calmness of the argmin mapping in terms of the
level set mapping (with respect to the objective function) and a special
supremum function. We also estimate the H\"older calmness modulus of the argmin
mapping in the framework of linear programming.Comment: 25 page
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