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2008

Characterizations of Lojasiewicz inequalities and applications

Bolte, Jerome, Daniilidis, Aris, Ley, Olivier et al.

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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 $-\partial f$ are strongly linked to this inequality.

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