Some New Characterizations of Intrinsic Transversality in Hilbert Spaces

Abstract

Motivated by a number of research questions concerning transversality-type properties of pairs of sets recently raised by Ioffe [21] and Kruger [31], this paper reports several new characterizations of the intrinsic transversality property in Hilbert spaces. Our dual space results clarify the picture of intrinsic transversality, its variants and the only existing sufficient dual condition for subtransversality, and actually unify them. New primal space characterizations of the intrinsic transversality which is originally a dual space condition lead to new understanding of the property in terms of primal space elements for the first time. As a consequence, the obtained analysis allows us to address a number of research questions asked by the two aforementioned researchers about the intrinsic transversality property in the Hilbert space setting. Mathematics Subject Classification IntroductionTransversality and subtransversality are the two important properties of collections of sets which reflect the mutual arrangement of the sets around the reference point in normed spaces. These properties are widely known as constraint qualification conditions in optimization and variational analysis for formulating optimality conditions [21,44,46,54] and calculus rules for subdifferentials, normal cones and coderivatives [18-20, 22, 32, 33, 44, 46], and as key ingredients for establishing sufficient and/

Transversality, Regularity and Error Bounds in Variational Analysis and Optimisation

“…The quantitative relations between transversality and 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 are discussed. These results are presented in recent publications [1, 3–6, 14].…”

“…The quantitative relations between transversality and 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 are discussed. These results are presented in recent publications [1,[3][4][5][6]14].We study theoretical and applied aspects of a new property called 'linear semitransversality of collections of set-valued mappings' in metric spaces [9]. The property can be seen as a generalisation of the conventional semitransversality of collections of sets and the negation of the corresponding stationarity which is a weaker property than the extremality of collections of set-valued mappings [12].…”