247 research outputs found
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
Quantitative Stability and Optimality Conditions in Convex Semi-Infinite and Infinite Programming
This paper concerns parameterized convex infinite (or semi-infinite)
inequality systems whose decision variables run over general
infinite-dimensional Banach (resp. finite-dimensional) spaces and that are
indexed by an arbitrary fixed set T . Parameter perturbations on the right-hand
side of the inequalities are measurable and bounded, and thus the natural
parameter space is . Based on advanced variational analysis, we
derive a precise formula for computing the exact Lipschitzian bound of the
feasible solution map, which involves only the system data, and then show that
this exact bound agrees with the coderivative norm of the aforementioned
mapping. On one hand, in this way we extend to the convex setting the results
of [4] developed in the linear framework under the boundedness assumption on
the system coefficients. On the other hand, in the case when the decision space
is reflexive, we succeed to remove this boundedness assumption in the general
convex case, establishing therefore results new even for linear infinite and
semi-infinite systems. The last part of the paper provides verifiable necessary
optimality conditions for infinite and semi-infinite programs with convex
inequality constraints and general nonsmooth and nonconvex objectives. In this
way we extend the corresponding results of [5] obtained for programs with
linear infinite inequality constraints
Constraint Qualifications and Optimality Conditions for Nonconvex Semi-Infinite and Infinite Programs
The paper concerns the study of new classes of nonlinear and nonconvex
optimization problems of the so-called infinite programming that are generally
defined on infinite-dimensional spaces of decision variables and contain
infinitely many of equality and inequality constraints with arbitrary (may not
be compact) index sets. These problems reduce to semi-infinite programs in the
case of finite-dimensional spaces of decision variables. We extend the
classical Mangasarian-Fromovitz and Farkas-Minkowski constraint qualifications
to such infinite and semi-infinite programs. The new qualification conditions
are used for efficient computing the appropriate normal cones to sets of
feasible solutions for these programs by employing advanced tools of
variational analysis and generalized differentiation. In the further
development we derive first-order necessary optimality conditions for infinite
and semi-infinite programs, which are new in both finite-dimensional and
infinite-dimensional settings.Comment: 28 page
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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