Clarke Subgradients of Stratifiable Functions

Bolte, Jérôme; Daniilidis, Aris; Lewis, Adrian S.; Shiota, Masahiro

doi:10.1137/060670080

Cited by 333 publications

(368 citation statements)

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“…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]. References on functions definable in an o-minimal structure are [26,30].…”

Section: Kurdyka-lojasiewicz Inequality: the Nonsmooth Casementioning

confidence: 99%

“…An important class is given by functions definable in an o-minimal structure. The monographs [26,30] are good references on o-minimal structures; concerning Kurdyka-Lojasiewicz inequalities in this context the reader is referred to [38,17]. Functions definable in o-minimal structures or functions whose graphs are locally definable are often called tame functions.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, our central assumption for the study of such algorithms is that the function f satisfies the (nonsmooth) Kurdyka-Lojasiewicz inequality, which means, roughly speaking, that the functions under consideration are sharp up to a reparametrization (see Section 2.2). The reader is referred to [44,45,38] for the smooth cases, and to [15,17] for nonsmooth inequalities. Kurdyka-Lojasiewicz inequalities have been successfully used to analyze various types of asymptotic behavior: gradient-like systems [15,34,35,39], PDE [55,22], gradient methods [1,48], proximal methods [3], projection methods or alternating methods [5,14].…”

Section: Introductionmentioning

confidence: 99%

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Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

2011

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To cite this version:Hedy Attouch, Jérôme Bolte, Benar Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program., Ser. A, 2011, 137 (1), pp.91-124. <10.1007/s10107-011-0484-9>. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods , 15, 2010; revised: July, 21, 2011 Abstract In view of the minimization of a nonsmooth nonconvex function f , we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka-Lojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward-backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss-Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.2010 Mathematics Subject Classification: 34G25, 47J25, 47J30, 47J35, 49M15, 49M37, 65K15, 90C25, 90C53.

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Section: Kurdyka-lojasiewicz Inequality: the Nonsmooth Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

2011

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“…Other fields of application of (1) are nonconvex optimization and nonsmooth analysis. This was one of the motivations for the nonsmooth K L-inequalities developed in [8,9]. Due to its considerable impact on several fields of applied mathematics: minimization and algorithms [1,5,8,39], asymptotic theory of differential inclusions [48], neural networks [28], complexity theory [47] (see [47,Definition 3], where functions satisfying a K L-type inequality are called gradient dominated functions), and partial differential equations [51,35,30,31], we hereby tackle the problem of characterizing such inequalities in a nonsmooth infinite-dimensional setting and provide further clarifications for several application aspects.…”

Section: Introductionmentioning

confidence: 99%

“…a slice of level sets. Under a compactness assumption and a condition of Sard type (automatically satisfied in finite dimensions if f belongs to an o-minimal class), their lengths are uniformly bounded if and only if f satisfies the K L-inequality in its nonsmooth form (see [9]); that is, for all x ∈ [0 < f < r],…”

Section: Introductionmentioning

confidence: 99%

Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity

Bolte

Daniilidis

Ley

et al. 2009

Trans. Amer. Math. Soc.

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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, and tame geometry. This paper provides alternative characterizations of this type of inequality for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In the framework of metric spaces, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka-Lojasiewicz inequality) is related 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 KurdykaLojasiewicz 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 us in particular to establish a kind of asymptotic equivalence for 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 fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka-Lojasiewicz inequality.

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Topological Aspects of Nonsmooth Optimization

Shikhman

2012

Springer Optimization and Its Applications

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The goal of this paper is to compare alternative stationarity notions in structured nonsmooth optimization (SNO). Here, nonsmoothness is caused by complementarity, vanishing, orthogonality type, switching, or disjunctive constraints. On one side, we consider geometrically motivated notions of N -, N -, and N -stationarity in terms of Fréchet, Mordukhovich, and Clarke normal cones to the feasible set, respectively. On the other side, we advocate the notion of topologically relevant T-stationarity, which adequately captures the global structure of SNO. Our main findings say that (a) N -stationary points include all local minimizers; (b) Nstationary points, which are not N -stationary, correspond to the singular saddle points of first order; (c) T-stationary points, which are not N -stationary, correspond to the regular saddle points of first order; (d) N -stationary points, which are not T-stationary, are irrelevant for optimization purposes. Overall, a hierarchy of stationarity notions for SNO is established.

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Clarke Subgradients of Stratifiable Functions

Cited by 333 publications

References 18 publications

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity

Topological Aspects of Nonsmooth Optimization

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