252 research outputs found
Error Bounds and Metric Subregularity
Necessary and sufficient criteria for metric subregularity (or calmness) of
set-valued mappings between general metric or Banach spaces are treated in the
framework of the theory of error bounds for a special family of extended
real-valued functions of two variables. A classification scheme for the general
error bound and metric subregularity criteria is presented. The criteria are
formulated in terms of several kinds of primal and subdifferential slopes.Comment: 32 page
Metric regularity of composition set-valued mappings: metric setting and coderivative conditions
The paper concerns a new method to obtain a direct proof of the openness at
linear rate/metric regularity of composite set-valued maps on metric spaces by
the unification and refinement of several methods developed somehow separately
in several works of the authors. In fact, this work is a synthesis and a
precise specialization to a general situation of some techniques explored in
the last years in the literature. In turn, these techniques are based on
several important concepts (like error bounds, lower semicontinuous envelope of
a set-valued map, local composition stability of multifunctions) and allow us
to obtain two new proofs of a recent result having deep roots in the topic of
regularity of mappings. Moreover, we make clear the idea that it is possible to
use (co)derivative conditions as tools of proof for openness results in very
general situations.Comment: 29 page
Perturbation of error bounds
Our aim in the current article is to extend the developments in Kruger, Ngai
& Th\'era, SIAM J. Optim. 20(6), 3280-3296 (2010) and, more precisely, to
characterize, in the Banach space setting, the stability of the local and
global error bound property of inequalities determined by proper lower
semicontinuous under data perturbations. We propose new concepts of (arbitrary,
convex and linear) perturbations of the given function defining the system
under consideration, which turn out to be a useful tool in our analysis. The
characterizations of error bounds for families of perturbations can be
interpreted as estimates of the `radius of error bounds'. The definitions and
characterizations are illustrated by examples.Comment: 20 page
Error bound results for convex inequality systems via conjugate duality
The aim of this paper is to implement some new techniques, based on conjugate
duality in convex optimization, for proving the existence of global error
bounds for convex inequality systems. We deal first of all with systems
described via one convex inequality and extend the achieved results, by making
use of a celebrated scalarization function, to convex inequality systems
expressed by means of a general vector function. We also propose a second
approach for guaranteeing the existence of global error bounds of the latter,
which meanwhile sharpens the classical result of Robinson.Comment: 12 page
Nonlinear Metric Subregularity
In this article, we investigate nonlinear metric subregularity properties of
set-valued mappings between general metric or Banach spaces. We demonstrate
that these properties can be treated in the framework of the theory of (linear)
error bounds for extended real-valued functions of two variables developed in
A. Y. Kruger, Error bounds and metric subregularity, Optimization 64, 1 (2015)
49-79. Several primal and dual space local quantitative and qualitative
criteria of nonlinear metric subregularity are formulated. The relationships
between the criteria are established and illustrated.Comment: 26 pages. arXiv admin note: substantial text overlap with
arXiv:1411.6414, arXiv:1405.113
BCQ and Strong BCQ for Nonconvex Generalized Equations with Applications to Metric Subregularity
In this paper, based on basic constraint qualification (BCQ) and strong BCQ
for convex generalized equation, we are inspired to further discuss constraint
qualifications of BCQ and strong BCQ for nonconvex generalized equation and
then establish their various characterizations. As applications, we use these
constraint qualifications to study metric subregularity of nonconvex
generalized equation and provide necessary and/or sufficient conditions in
terms of constraint qualifications considered herein to ensure nonconvex
generalized equation having metric subregularity.Comment: 17 page
Preconditioned proximal point methods and notions of partial subregularity
Based on the needs of convergence proofs of preconditioned proximal point
methods, we introduce notions of partial strong submonotonicity and partial
(metric) subregularity of set-valued maps. We study relationships between these
two concepts, neither of which is generally weaker or stronger than the other
one. For our algorithmic purposes, the novel submonotonicity turns out to be
easier to employ than more conventional error bounds obtained from
subregularity. Using strong submonotonicity, we demonstrate the linear
convergence of the Primal-Dual Proximal splitting method to some strictly
complementary solutions of example problems from image processing and data
science. This is without the conventional assumption that all the objective
functions of the involved saddle point problem are strongly convex
On a strong covering property of multivalued mappings
In this paper, a strong variant for multivalued mappings of the well-known
property of openness at a linear rate is studied. Among other examples, a
simply characterized class of closed convex processes between Banach spaces,
which satisfies such a covering behaviour, is singled out. Equivalent
reformulations of this property and its stability under Lipschitz perturbations
are investigated in a metric space setting. Applications to the solvability of
set-valued inclusions and to the exact penalization of optimization problems
with set-inclusion constraints are discussed
Directional H\"older Metric Regularity
This paper sheds new light on regularity of multifunctions through various
characterizations of directional H\"older /Lipschitz metric regularity, which
are based on the concepts of slope and coderivative. By using these
characterizations, we show that directional H\"older /Lipschitz metric
regularity is stable, when the multifunction under consideration is perturbed
suitably. Applications of directional H\"older /Lipschitz metric regularity to
investigate the stability and the sensitivity analysis of parameterized
optimization problems are also discussed
Metric Regularity. Theory and Applications - a survey
Metric regularity has emerged during last 2-3 decades as one of the central
concepts of variational analysis. The roots of this concept go back to a circle
of fundamental regularity ideas of classical analysis embodied in such results
as the implicit function theorem, Banach open mapping theorem, theorems of
Lyusternik and Graves, on the one hand, and the Sard theorem and the Thom-Smale
transversality theory, on the other. The three principal themes that are in the
focus of attention are: (a) regularity criteria (containing quantitative
estimates for rates of regularity) including formal comparisons of their
relative power and precision; (b) stability problems relating to the effect of
perturbations of the mapping on its regularity properties, on the one hand, and
to solutions of equations, inclusions etc. on the other; (c) role of metric
regularity in analysis and optimization. All of them are studied at three
levels of generality: the general theory for (set-valued) mappings between
metric spaces is followed by a detailed study of Banach and finite dimensional
theories. There is a number of new results, both theoretical and relating to
applications, and some known results are supplied with new, usually simpler,
proofs
- …