Alexander Y. Kruger A general nonlinear regularity model for a set-valued mapping $F:X\times\R_+\rightrightarrows Y$, where $X$ and $Y$ are metric spaces, is considered using special iteration procedures, going back to Banach, Schauder, Lusternik and Graves. Namely, we revise the \emph{induction theorem} from Khanh, \emph{J. Math. Anal. Appl.}, 118 (1986) and employ it to obtain basic estimates for studying … Read more
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 … Read more
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 … Read more
Several primal and dual characterizations of regularity properties of collections of sets in normed linear spaces are discussed. Relationships between regularity properties of collections of sets and those of set-valued mappings are provided. CitationJOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS (2015) 164:41–67ArticleDownload View PDF
We further investigate the uniform regularity property of collections of sets via primal and dual characterizing constants. These constants play an important role in determining convergence rates of projection algorithms for solving feasibility problems. CitationPublished in Serdica Math. J. 39, 287–312 (2013) http://www.math.bas.bg/serdica/2013/2013-287-312.pdfArticleDownload View PDF
Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide … Read more
We consider lexicographic vector equilibrium problems in metric spaces. Sufficient conditions for a family of such problems to be (uniquely) well-posed at the reference point are established. As an application, we derive several results on well-posedness for a class of variational inequalities. CitationPublished in Constructive Nonsmooth Analysis and Related Topics (Vladimir Demyanov, Panos M. Pardalos, … Read more
For the classical nonlinear program two new relaxations of the Mangasarian-Fromovitz constraint qualification are discussed and their relationship with some standard constraint qualifications is examined. In particular, we establish the equivalence of one of these constraint qualifications with the recently suggested by Andreani et al. Constant rank of the subspace component constraint qualification. As an … Read more
This article aims to demonstrate how the definitions of slopes can be extended to multi-valued mappings between metric spaces and applied for characterizing metric regularity. Several kinds of local and nonlocal slopes are defined and several metric regularity properties for set-valued mappings between metric spaces are investigated. CitationPublished in Vietnam Journal of Mathematics 40:2&3(2012) 355-369. … Read more
The paper presents a general primal space classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several primal space derivative-like objects – slopes – are used to characterize the error bound property of extended-real-valued functions on metric sapces. CitationPublished in D. Klatte et al. (eds.), Operations Research Proceedings … Read more