Metric regularity, openness and Lipschitzian behavior of multifunctions

Penot, Jean-Paul

doi:10.1016/0362-546x(89)90083-7

Cited by 141 publications

(77 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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)|. 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…”

Section: Introductionmentioning

confidence: 81%

See 1 more Smart Citation

Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity

Bolte

Daniilidis

Ley

et al. 2009

Trans. Amer. Math. Soc.

277

307

View full text Add to dashboard Cite

Abstract. The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, and tame geometry. This paper provides alternative characterizations of this type of inequality for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In the framework of metric spaces, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka-Lojasiewicz inequality) is related to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by −∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the KurdykaLojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of f -and integrability conditions are given. In the convex case these results are significantly reinforced, allowing us in particular to establish a kind of asymptotic equivalence for discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C 2 function in R 2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka-Lojasiewicz inequality.

show abstract

Section: Introductionmentioning

confidence: 81%

“…We refer to Ioffe [32,33], Mordukhovich [45], Penot [49], Dontchev-LewisRockafellar [23], Dontchev-Quincampoix-Zlateva [24] and the references therein for historical comments, examples, other characterizations and applications of the notion of metric regularity.…”

Section: Definition 1 (Metric Regularity Of Multifunctions) Letmentioning

confidence: 99%

Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity

Bolte

Daniilidis

Ley

et al. 2009

Trans. Amer. Math. Soc.

277

307

View full text Add to dashboard Cite

show abstract

“…Several conditions using subdifferential operators or directional derivatives and ensuring the error bound in Banach spaces have been established, for example, in [16], [35], [49], [46], [58]. Recently, Azé [4], Azé & Corvellec [7] have used the so-called strong slope introduced by De Giorgi, Marino & Tosques in [19] to prove criteria for error bounds in complete metric spaces.…”

Section: Characterizations Of Error Bounds For Parametric Inequality mentioning

confidence: 99%

Implicit multifunction theorems in complete metric spaces

2013

View full text Add to dashboard Cite

In this paper, we establish some new characterizations of the metric regularity of implicit multifunctions in complete metric spaces by using the lower semicontinuous envelopes of the distance functions for set-valued mappings. Through these new characterizations it is possible to investigate implicit multifunction theorems based on coderivatives and on contingent derivatives as well as the perturbation stability of implicit multifunctions.Mathematics Subject Classification: 49J52, 49J53, 90C30.

show abstract

“…According to the long history of metric regularity there is an abundant literature on conditions ensuring this property. We refer the reader to the basic monographs [2,3,4,5,6], to the excellent survey of A. Ioffe [7] (in preparation) and to some (non exhaustives) references [1,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25].…”

mentioning

confidence: 99%

“…Apart from the study of the usual (Lipschitz) metric regularity, Hölder metric regularity or more generally nonlinear metric regularity have been studied over the years 1980 − 1990s by several authors, including for example Borwein and Zhuang [1], Frankowska [16], Penot [25], and recently, for instance, Frankowska and Quincampoix [17], Ioffe [26], Li and Mordukhovich [27], Oyang and Mordukovhich [28].…”

mentioning

confidence: 99%

Directional Hölder Metric Regularity

Huynh

Nguyen

Théra

2015

J Optim Theory Appl

View full text Add to dashboard Cite

This paper sheds new light on regularity of multifunctions through various characterizations of directional Hölder/Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional Hölder/Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directional Hölder/Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed.

show abstract

Metric regularity, openness and Lipschitzian behavior of multifunctions

Cited by 141 publications

References 20 publications

Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity

Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity

Implicit multifunction theorems in complete metric spaces

Directional Hölder Metric Regularity

Contact Info

Product

Resources

About