Characterizations of Lojasiewicz inequalities and applications

By Jérôme Bolte, Aris Daniilidis, Olivier Ley, Laurent Mazet. 2008

37 citations recorded by the source corpus.

The classical Lojasiewicz inequality and its extensions for partial differential equa- tion 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 the- ory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicon- tinuous 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 inequali- ty) 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 subgra- dient 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 establish the asymptotic equivalence of 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

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Hédy Attouch
2. Eric P. Xing
3. Jérôme Bolte
4. Yaoliang Yu

In Computational Theory and Mathematics

1. Hédy Attouch
2. Patrick Redont
3. Jérôme Bolte
4. Antoine Soubeyran
5. Eric P. Xing
6. Yaoliang Yu
7. Yi Zhou
8. Wei Dai

In Mathematical Physics

1. Hédy Attouch
2. Jérôme Bolte
3. Patrick Redont
4. Eric P. Xing
5. Yi Zhou
6. Yaoliang Yu
7. Wei Dai
8. Yingbin Liang

Citations in the corpus

• Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality (2008, background): “More recently Bolte, Daniilidis, Lewis and Shiota have extended the Kurdyka-Lojasiewicz inequality to nonsmooth functions in the subanalytic and o-minimal setting [16, 17, 18]; see also [15].”
• On Convergence of Model Parallel Proximal Gradient Algorithm for Stale Synchronous Parallel System (2016, methodology): “The KL inequality (7) is an important tool to bound the trajectory length of a dynamical system (see [10, 20] and the references therein for some historic developments).”