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

Tangential Extremal Principles for Finite and Infinite Systems of Sets, II: Applications to Semi-infinite and Multiobjective Optimization

This paper contains selected applications of the new tangential extremal principles and related results developed in Part I to calculus rules for infinite intersections of sets and optimality conditions for problems of semi-infinite programming and multiobjective optimization with countable constraint

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

About uniform regularity of collections of sets

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

The strong conical hull intersection property for systems of closed convex sets.

Pong Ting Kei.Thesis (M.Phil.)--Chinese University of Hong Kong, 2006.Includes bibliographical references (leaves 79-82).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.5Chapter 2 --- Preliminary --- p.7Chapter 2.1 --- Introduction --- p.7Chapter 2.2 --- Notations --- p.7Chapter 2.3 --- On properties of Normal Cones --- p.9Chapter 2.4 --- Polar Calculus --- p.13Chapter 2.5 --- Notions of Relative Interior --- p.17Chapter 2.6 --- Properties of Minkowski functional --- p.18Chapter 2.7 --- Properties of Epigraphs --- p.19Chapter 3 --- The Strong Conical Hull Intersection Property (Strong CHIP): Definition and Some Properties --- p.22Chapter 3.1 --- Introduction --- p.22Chapter 3.2 --- Definition of the strong CHIP --- p.24Chapter 3.3 --- Relationship between the strong CHIP and projections onto sets --- p.26Chapter 3.4 --- Relationship between the strong CHIP and the Basic Constraint Qualifications (BCQ) --- p.35Chapter 3.5 --- The strong CHIP of extremal subsets --- p.42Chapter 4 --- Sufficient Conditions for the Strong CHIP --- p.46Chapter 4.1 --- Introduction --- p.46Chapter 4.2 --- ̐ưجI̐ưجis finite --- p.47Chapter 4.2.1 --- Interior point conditions --- p.47Chapter 4.2.2 --- Boundedly linear regularity --- p.52Chapter 4.2.3 --- Epi-sum --- p.54Chapter 4.3 --- ̐ưجI̐ưجis infinite --- p.56Chapter 4.3.1 --- A Sum Rule --- p.57Chapter 4.3.2 --- The C-Extended Minkowski Functional --- p.58Chapter 4.3.3 --- Relative Interior Point Conditions --- p.62Chapter 4.3.4 --- Bounded Linear Regularity --- p.68Chapter 5 --- "The SECQ, Linear Regularity and the Strong CHIP for Infinite System of Closed Convex Sets in Normed Linear Spaces" --- p.69Chapter 5.1 --- Introduction --- p.69Chapter 5.2 --- The strong CHIP and the SECQ --- p.71Chapter 5.3 --- Linear regularity and the SECQ --- p.73Chapter 5.4 --- Interior-point conditions and the SECQ --- p.76Bibliography --- p.7