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Characterizations of Lojasiewicz inequalities and applications: subgradient flows, talweg, convexity

By Jérôme Bolte, Aris Daniilidis, Olivier Ley and Laurent Mazet


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, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka-Lojasiewicz inequality) relates 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 Kurdyka-Lojasiewicz 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 in particular to establis

Year: 2013
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