On semiregularity of mappings

Abstract

There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature. We provide a self-contained material gathering and extending the existing theory on the topic. We demonstrate a clear relationship with other regularity properties, for example, the equivalence with the so-called openness with a linear rate at the reference point is shown. In particular cases, we derive necessary and/or sufficient conditions of both primal and dual type. We illustrate the importance of semiregularity in the convergence analysis of an inexact Newton-type scheme for generalized equations with not necessarily differentiable single-valued part.

Geometric and Metric Characterizations of Transversality Properties

“…In fact, rg[F](x,ȳ) coincides with the modulus of surjection [17]. Unlike its more famous siblings, the first property in Definition 2 has only recently started attracting attention of researchers; see [1,8,21].…”

Calmness and Calculus: Two Basic Patterns

“…Moreover, the estimates for primal and directional limiting objects were recently shown to hold under inner calmness and inner calmness* in [4,6]. While inner calmness* was newly defined in [4], inner calmness can be found under other names, such as, e.g., Lipschitz lower semicontinuity, see [33,34], or recession with linear rate, see [8,28], as well. We refer to [8] for a comprehensive overview of this and related notions.…”

“…While inner calmness* was newly defined in [4], inner calmness can be found under other names, such as, e.g., Lipschitz lower semicontinuity, see [33,34], or recession with linear rate, see [8,28], as well. We refer to [8] for a comprehensive overview of this and related notions. Let us also mention the stronger concept of Lipschitz lower semicontinuity*, recently introduced in [9] and motivated by a relaxation of the Aubin property from earlier works of Klatte [31,32].…”

Directional Metric Pseudo Subregularity of Set-valued Mappings: a General Model

“…By the time and demand of use and applications, variants of this property have emerged suitable to practical problems. Weaker/stronger versions: calmness, (strong) (Hölder) metric sub/regularity, semiregularity or equivalent versions: pseudo Lipschitz, linear openness were studied and have proved to have an important role in various applications in Mathematics, especially in Variational Analysis and Optimization [5,12,13,[15][16][17]26], ... Another direction in this line is to build directional models for these objects as recently proposed by Arutyunov-Avakov-Izmailov [1], Gfrerer [8], Ngai-Théra [20], Ngai-Tron-Théra [22], Ngai-Tron-Tinh [23]. Characterizations of these concepts have been established and successfully applied to study optimality conditions for mathematical programs, for calculating tangent cones,...This notion of directional regularity is an extension of an earlier notion used by Bonnans and Shapiro [3] to study sensitivity analysis.…”