### Abstract

Given a real-analytic function f : ℝ n → ℝ and a critical point a ε ℝ n, the Łojasiewicz inequality asserts that there exists θ ε [1/2, 1) such that the function |f - f(a)| θ ||∇f|| _1 remains bounded around a. In this paper, we extend the above result to a wide class of nonsmooth functions (that possibly admit the value +∞), by establishing an analogous inequality in which the derivative ∇ f(x) can be replaced by any element x* of the subdifferential ∂f(x) of f. Like its smooth version, this result provides new insights into the convergence aspects of subgradient-type dynamical systems. Provided that the function f is sufficiently regular (for instance, convex or lower-C 2), the bounded trajectories of the corresponding subgradient dynamical system can be shown to be of finite length. Explicit estimates of the rate of convergence are also derived. © 2007 Society for Industrial and Applied Mathematics.

Original language | English |
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Pages (from-to) | 1205-1223 |

Journal | SIAM Journal on Optimization |

Volume | 17 |

DOIs | |

Publication status | Published - 1 Dec 2006 |

### Keywords

- Descent method
- Dynamical system
- Nonsmooth analysis
- Subanalytic function
- Subdifferential
- ŁOjasiewicz inequality

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## Cite this

Bolte, J., Daniilidis, A., & Lewis, A. (2006). The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems.

*SIAM Journal on Optimization*,*17*, 1205-1223. https://doi.org/10.1137/050644641