Some Remarks on the Class of Continuous (Semi-) Strictly Quasiconvex Functions

Daniilidis, Aris; Ramos, Yboon Garcìa

doi:10.1007/s10957-007-9182-4

Cited by 10 publications

(7 citation statements)

References 26 publications

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“…Theorem 1 can be extended to the case where R is a semi-strictly quasi-convex function. A function R is said to be semi-strictly quasi-convex [14] if it is quasi-convex and if…”

Section: Remark 3 (Gauge Functions or Semi-norms)mentioning

confidence: 99%

On Representer Theorems and Convex Regularization

Boyer¹,

Chambolle²,

Castro³

et al. 2019

SIAM J. Optim.

130

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We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and elements of the extreme rays of the regularizer level sets. An extension to a broader class of quasiconvex regularizers is also discussed. As a side result, we characterize the minimizers of the total gradient variation, which was still an unresolved problem.

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“…Theorem 1 can be extended to the case where R is a semi-strictly quasi-convex function. A function R is said to be semi-strictly quasi-convex [14] if it is quasi-convex and if…”

Section: Remark 3 (Gauge Functions or Semi-norms)mentioning

confidence: 99%

On Representer Theorems and Convex Regularization

Boyer¹,

Chambolle²,

Castro³

et al. 2019

SIAM J. Optim.

130

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“…The above definition is thus equivalent to [6,Definition (6.5)]. to assume that f is semistrictly quasiconvex, see for instance [5] for the relevant definition and characterizations.…”

Section: 1mentioning

confidence: 99%

Rectifiability of Self-Contracted Curves in the Euclidean Space and Applications

et al. 2013

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Abstract. It is hereby established that, in Euclidean spaces of finite dimension, bounded selfcontracted curves have finite length. This extends the main result of [6] concerning continuous planar self-contracted curves to any dimension, and dispenses entirely with the continuity requirement. The proof borrows heavily from a geometric idea of [13] employed for the study of regular enough curves, and can be seen as a nonsmooth adaptation of the latter, albeit a nontrivial one. Applications to continuous and discrete dynamical systems are discussed: continuous self-contracted curves appear as generalized solutions of nonsmooth convex foliation systems, recovering a hidden regularity after reparameterization, as consequence of our main result. In the discrete case, proximal sequences (obtained through implicit discretization of a gradient system) give rise to polygonal self-contracted curves. This yields a straightforward proof for the convergence of the exact proximal algorithm, under any choice of parameters.

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“…Actually, it follows from [9, Theorem 3.1] that the sublevel sets of a continuous quasiconvex coercive function f define a convex foliation if and only if the function is semi-strictly quasiconvex. (We refer to [9] for the exact definition and basic properties of semi-strictly quasiconvex functions.) (ii) Let f : R 2 → R be a coercive C 1,1 quasiconvex function and γ : [0, +∞) → R 2 be the solution of (6.1).…”

Section: Subgradient Dynamical Systems -Convex Casementioning

confidence: 99%

Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions

Daniilidis

Ley

Sabourau

2010

Journal de Mathématiques Pures et Appliquées

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International audienceWe hereby introduce and study the notion of self-contracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded self-contracted planar curves have a finite length. We also give an example of a convex function defined in the plane whose gradient orbits spiral infinitely many times around the unique minimum of the function

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Some Remarks on the Class of Continuous (Semi-) Strictly Quasiconvex Functions

Cited by 10 publications

References 26 publications

On Representer Theorems and Convex Regularization

On Representer Theorems and Convex Regularization

Rectifiability of Self-Contracted Curves in the Euclidean Space and Applications

Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions

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