Characterizations of error bounds for lower semicontinuous functions on metric spaces

Abstract

Abstract.Refining the variational method introduced in Azé et al. [Nonlinear Anal. 49 (2002) 643-670], we give characterizations of the existence of so-called global and local error bounds, for lower semicontinuous functions defined on complete metric spaces. We thus provide a systematic and synthetic approach to the subject, emphasizing the special case of convex functions defined on arbitrary Banach spaces (refining the abstract part of Azé and Corvellec [SIAM J. Optim. 12 (2002) 913-927], and the characterization of the local metric regularity of closed-graph multifunctions between complete metric spaces.Mathematics Subject Classification. 49J52, 90C26, 90C25, 49J53.

Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity

“…(Strong slope) Let us recall from [22] (see also [32], [6]) the notion of a strong slope defined for every x ∈ dom f as follows:…”

“…This fact is strongly connected to famous classical results in this area (see [23,45,32,49] for example) and in particular to the notion of ρ-metric regularity introduced in [32] by Ioffe. Using results on global error-bounds of Ioffe [32,33,34] (see also Azé-Corvellec [6]) we show that some global forms of the K L-inequality, as well as metric regularity, are both equivalent to the "Lipschitz continuity" of the sublevel mapping…”

“…The underlying space H is only assumed to be a complete metric space (without any linear structure), the function f : H → R∪{+∞} is lower semicontinuous and possibly real-extended valued and the notion of a gradient is replaced by the variational notion of a strong-slope [22,6]. Indeed, introducing the multivalued mapping F (x) = [f (x), +∞) (whose graph is the epigraph of f ), the K L-inequality (1) appears to be equivalent to the metric regularity of F : H ⇒ R on an adequate set, where R is endowed with the metric d ϕ (r, s) = |ϕ(r) − ϕ(s)|.…”

Directional Hölder Metric Regularity

“…Recall from [39], (see also [40] and the discussion in [38]) that the local slope and the nonlocal slope (see, e.g., [41]) of the function f at x ∈ dom f , are the quantities denoted respectively by |∇ f |(x) and |Γ f |(x). Using the notation…”

Implicit multifunction theorems in complete metric spaces

“…Our main objective in this paper is to use the theory of error bounds to study the metric regularity of implicit multifunctions. The approach based on error bounds to investigate the metric regularity has been recently used in [6] and in [46] to study implicit multifunctions in smooth spaces. Especially in the survey paper [5], it was shown that this approach is powerful to provide a unified theory of the metric regularity.…”