The [barred L]ojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems

By Jérôme Bolte, Aris Daniilidis, A. S. Lewis. SIAM Journal on Optimization, 2006

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Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/049273b64f04d0ba9cb69f7c134609b865a5c180

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A nonsmooth Morse–Sard theorem for subanalytic functions (2006, methodology): “Proof. (i)=)(ii). It follows from [ 4, Theorem 3.5 ].”
Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods (2013, methodology): “Among real-extended-valued lower-semicontinuous functions, typical KL functions are semi-algebraic functions or more generally functions definable in an o-minimal structure, see [15, 16, 17].”
Projection-Free Optimization on Uniformly Convex Sets (2020, background): “We say that f satisﬁes a ( µ, θ ) -Hölderian Error Bound when there exists θ (HEB) When the function f is subanalytic, (HEB) is known to hold generically (Lojasiewicz, 1965; Kurdyka, 1998; Bolte et al., 2007).”
The Frank-Wolfe Algorithm: A Short Introduction (2023, neutral): “…[46], Zhang et al. [86], Chen et al. [20], Garber and Kretzu [39], Kerdreux et al. [51], Zhang et al. [87]) and many more (see, e.g., Bolte et al. [6], Clarkson [22], Pierucci et al. [70], Harchaoui et al. [44], Wang et al. [81], Cheung and Li [21], Ravi et al. [72], Hazan and Minasyan [47],…”
On Nonconvex Decentralized Gradient Descent (2016, neutral): “Definition 3 (Kurdyka-Łojasiewicz (KŁ) property [2], [7], [34]): A function h : R → R ∪ {+∞} has the KŁ property at x∗ ∈ dom(∂h) if there exist η ∈ (0,+∞], a neighborhood U of x∗, and a continuous concave function φ : [0, η)→ R+ such that: i) φ(0) = 0 and φ is differentiable on (0, η); ii) for all s”
New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors (2015, result): “Note that much of our study on error bounds is in the spirit of [35, 36, 27, 28] being largely motivated by the recent work on nonsmooth extensions of Lojasiewicz’s inequality initiated in [3].”
Optimality and Approximation with Policy Gradient Methods in Markov Decision Processes (2019, result): “…domination property if for all θ ∈ Θ, f(θ⋆)− f(θ) = O(G(θ)), where θ⋆ ∈ argmaxθ′∈Θ f(θ′) and where G(θ) is some suitable scalar notion of first-order stationarity, which can be considered a measure of how large the gradient is (see [Karimi et al., 2016, Bolte et al., 2007, Attouch et al., 2010]).”
A Globally Convergent Algorithm for Nonconvex Optimization Based on Block Coordinate Update (2014, background): “4) was extended to nonsmooth sub-analytic functions [10].”
A Block Coordinate Descent Method for Regularized Multiconvex Optimization with Applications to Nonnegative Tensor Factorization and Completion (2013, background): “Recently, the KL inequality was extended to nonsmooth sub-analytic functions [13].”
Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality (2008, background): “Then f complies with (14) at point x̄ (for φ(s) = r c 1 r s 1 r ) on U ∩ [min f < f < min f + η] (see [16]).”
Modern regularization methods for inverse problems (2018, background): “In a finite dimensional setting U = Rn it is now sufficient to assume that E + J is a KurdykaLojasiewicz (KL) function [270, 255, 45] in order to show that the iterates (85) converge globally to a critical point of E + J .”
Forward–backward quasi-Newton methods for nonsmooth optimization problems (2016, neutral): “In case where φ is subanalytic, the desingularizing function can be taken of the form ψ(s) = σ s1−θ , for σ > 0 and θ ∈ [0, 1) [52].”
On the convergence of the proximal algorithm for nonsmooth functions involving analytic features (2008, methodology): “Our special interest for the proximal method for functions involving analytic features comes from the recent developments concerning the convergence of the steepest descent method by Simon [30], Haraux [14] and Bolte-DaniilidisLewis [7].”
Proximal alternating linearized minimization for nonconvex and nonsmooth problems (2013, background): “Then σ satisfies the KL property at the point x̄ for φ (s) = r c− 1 r s 1 r on the set U ∩ [minσ < σ < minσ + η] (see, for more details, [15,16]).”
From error bounds to the complexity of first-order descent methods for convex functions (2015, background): “3 KurdykaLojasiewicz inequality In this subsection, we present the nonsmooth KurdykaLojasiewicz inequality introduced in [17] (see also [19, 18], and the fundamental works [43, 38]).”
On Convergence of Model Parallel Proximal Gradient Algorithm for Stale Synchronous Parallel System (2016, background): “Then, inspired by the recent KurdykaLojasiewicz (K L) inequality [3, 5, 9, 11, 20], we further prove in Theorem 2 that the whole sequence of msPG in fact converges to a critical point, under mild technical assumptions that we verify for many familiar examples.”