About Regularity of Collections of Sets

Kruger, Alexander Y.

doi:10.1007/s11228-006-0014-8

Cited by 78 publications

(140 citation statements)

References 31 publications

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“…and it is equivalent to our previous definition (see [28,Corollary 2], for example). We also note that this condition appears throughout variationalanalytic theory.…”

Section: Notation and Definitionsmentioning

confidence: 96%

“…(In [28] this property is called "strong regularity".) By considering the case x = z =x, we see that linear regularity implies thatx is not locally extremal.…”

Section: Notation and Definitionsmentioning

confidence: 99%

“…A concise summary of the relationships between a variety of regular intersection properties and metric regularity appears in [28]. We summarize the relevant ideas here.…”

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confidence: 99%

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Local Linear Convergence for Alternating and Averaged Nonconvex Projections

2008

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The idea of a finite collection of closed sets having "linearly regular intersection" at a point is crucial in variational analysis. This central theoretical condition also has striking algorithmic consequences: in the case of two sets, one of which satisfies a further regularity condition (convexity or smoothness for example), we prove that von Neumann's method of "alternating projections" converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having linearly regular intersection at some point, the method of "averaged projections" converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly.

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“…and it is equivalent to our previous definition (see [28,Corollary 2], for example). We also note that this condition appears throughout variationalanalytic theory.…”

Section: Notation and Definitionsmentioning

confidence: 96%

“…(In [28] this property is called "strong regularity".) By considering the case x = z =x, we see that linear regularity implies thatx is not locally extremal.…”

Section: Notation and Definitionsmentioning

confidence: 99%

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Local Linear Convergence for Alternating and Averaged Nonconvex Projections

2008

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“…Assume that the restriction of f to its domain is a continuous function. If a sequence (x k ) k∈N generated by Algorithm 2 (or by (36), (37) and (39)) is bounded, then it converges to some critical pointx of f . Moreover the sequence (x k ) k∈N has a finite length, i.e.…”

Section: Theorem 42 (Inexact Proximal Algorithm)mentioning

confidence: 99%

“…To this end, we will use the following result [42, Proposition 8.5] (based itself on a characterization given in [37]). …”

Section: Examplesmentioning

confidence: 99%

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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