664 research outputs found
Tangential Extremal Principles for Finite and Infinite Systems of Sets, I: Basic Theory
In this paper we develop new extremal principles in variational analysis that
deal with finite and infinite systems of convex and nonconvex sets. The results
obtained, unified under the name of tangential extremal principles, combine
primal and dual approaches to the study of variational systems being in fact
first extremal principles applied to infinite systems of sets. The first part
of the paper concerns the basic theory of tangential extremal principles while
the second part presents applications to problems of semi-infinite programming
and multiobjective optimization
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
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
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
Rated Extremal Principles for Finite and Infinite Systems
In this paper we introduce new notions of local extremality for finite and
infinite systems of closed sets and establish the corresponding extremal
principles for them called here rated extremal principles. These developments
are in the core geometric theory of variational analysis. We present their
applications to calculus and optimality conditions for problems with infinitely
many constraints
Some aspects of variational inequalities
AbstractIn this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence, uniqueness, computational properties, various generalizations, sensitivity analysis and their applications. We also propose some open problems with sufficient information and references, so that someone may attempt solution(s) in his/her area of special interest. We also include some new results, which we have recently obtained
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