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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity


Authors: Jérôme Bolte, Aris Daniilidis, Olivier Ley and Laurent Mazet
Journal: Trans. Amer. Math. Soc. 362 (2010), 3319-3363
MSC (2010): Primary 26D10; Secondary 03C64, 37N40, 49J52, 65K10
Published electronically: December 22, 2009
MathSciNet review: 2592958
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Abstract: The classical Łojasiewicz 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 Łojasiewicz inequality (hereby called the Kurdyka-Łojasiewicz 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 $ -\partial 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 Kurdyka-Łojasiewicz 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 $ \mathbb{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-Łojasiewicz inequality.


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Additional Information

Jérôme Bolte
Affiliation: UPMC Université Paris 06 - Équipe Combinatoire et Optimisation (UMR 7090), Case 189, Université Pierre et Marie Curie, 4 Place Jussieu, F–75252 Paris Cedex 05, France
Email: bolte@math.jussieu.fr

Aris Daniilidis
Affiliation: Departament de Matemàtiques, C1/308, Universitat Autònoma de Barcelona, E–08193 Bellaterra (Cerdanyola del Vallès), Spain
Email: arisd@mat.uab.es

Olivier Ley
Affiliation: Laboratoire de Mathématiques et Physique Théorique (CNRS UMR 6083), Fédération Denis Poisson, Faculté des Sciences et Techniques, Université François Rabelais, Parc de Grandmont, F–37200 Tours, France
Address at time of publication: IRMAR (CNRS UMR 6625) INSA de Rennes, 20 avenue des buttes de Coesmes, F-35708 Rennes Cedex 7, France
Email: ley@lmpt.univ-tours.fr, olivier.ley@univ-rennes1.fr

Laurent Mazet
Affiliation: Université Paris-Est, Laboratoire d’Analyse et Mathématiques Appliquées, UMR 8050, UFR des Sciences et Technologie, Département de Mathématiques, 61 avenue du Général de Gaulle 94010 Créteil cedex, France
Email: laurent.mazet@univ-paris12.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-09-05048-X
PII: S 0002-9947(09)05048-X
Keywords: \L ojasiewicz inequality, gradient inequalities, metric regularity, subgradient curve, talweg, gradient method, convex functions, global convergence, proximal method.
Received by editor(s): February 7, 2008
Received by editor(s) in revised form: March 11, 2009
Published electronically: December 22, 2009
Article copyright: © Copyright 2009 American Mathematical Society