Metric regularity and Lipschitzian stability of parametric variational systems

Abstract

Abstract. The paper concerns the study of variational systems described by parameterized generalized equations/variational conditions important for many aspects of nonlinear analysis, optimization, and their applications. Focusing on the fundamental properties of metric regularity and Lipschitzian stability, we establish various qualitative and quantitative relationships between these properties for multivalued parts/fields of parametric generalized equations and the corresponding solution maps for them in the framework of arbitrary Banach spaces of decision and parameter variables.

Higher-order metric subregularity and its applications

“…where f is single-valued while Q(y) = ∂ϕ(y) is a set-valued mapping of the subdifferential/normal cone type generated by nonsmooth functions; see [23] and also [2][3][4]15,17,20,21,29] for more details and further results in this direction. However, this phenomenon does not appear if metric regularity is replaced by a weaker property of metric subregularity of F at (x,ȳ) defined by…”

On Some Regularity Properties in Variational Analysis

“…The aim of this work is, in other terms, to investigate relationships between certain regularity properties of the solution map S and those of the base and of the field in (PGE). In such sense the present paper is rather close in the spirit to [1]. Nonetheless, while in the latter the major emphasis is put on relationships between S and the field Q, here the focus is on the properties inherited by S from the base f , with the field playing instead a perturbation role (at least, this is a possible reading key of Theorem 3.2 below).…”

“…Such failure phenomenon has strict connections with the generic ill-posedness of inverse problems, which has been observed for instance in the parameter identification problems (for an extended discussion, see [1]). Stimulated by such kind of results, in the present paper a study is undertaken of conditions on parametrized generalized equation data, which are able to ensure that metric regularity for the corresponding variational systems do actually hold.…”

“…Within such context, one of the latest achievements on the theme of Lipschitian behaviour is the fact that metric regularity, which is a remarkable property of set-valued maps playing a crucial role in many aspects of variational analysis and its applications, surprisingly fails to hold for certain general classes of variational systems (see [1,7,15]). Such failure phenomenon has strict connections with the generic ill-posedness of inverse problems, which has been observed for instance in the parameter identification problems (for an extended discussion, see [1]).…”

“…Another distinguishing feature relies on the respective techniques of analysis. While in [1] an approach is employed, which is based on an enhanced iterative process of Lyusternik-Graves type and on a reformulation of the contraction principle for set-valued maps, here the main analysis tool will be the Ekeland variational principle (see [6]). …”