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

被引：27

作者：

Daniilidis, Aris ^{[1
,2
]}

Ley, Olivier ^{[2
,3
]}

Sabourau, Stephane ^{[2
]}

机构：

[1] Univ Autonoma Barcelona, Dept Matemat, E-08193 Bellaterra, Cerdanyola Vall, Spain

[2] Univ Tours, Lab Math & Phys Theor, CNRS, Federat Rech Denis Poisson FR 2964,UMR 6083, F-37400 Tours, France

[3] INSA Rennes, IRMAR, CNRS, UMR 6625, F-35708 Rennes, France

来源：

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES | 2010年 / 94卷 / 02期

关键词：

Planar dynamical system; Gradient trajectory; Convex function; Convex foliation; Lojasiewicz inequality; O-MINIMAL STRUCTURES; HILBERT-SPACE; CONVERGENCE;

D O I：

10.1016/j.matpur.2010.03.007

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We 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 minimizer of the function. (C) 2010 Elsevier Masson SAS. All rights reserved.

引用

页码：183 / 199

页数：17

共 17 条

[1] SELF-CONTRACTED CURVES ARE GRADIENT FLOWS OF CONVEX FUNCTIONS
Durand-cartagena, Estibalitz
Lemenant, Antoine
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2019, 147 (06) : 2517 - 2531
[2] Self-contracted curves in Riemannian manifolds
Daniilidis, Aris
Deville, Robert
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Rifford, Ludovic
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2018, 457 (02) : 1333 - 1352
[3] Self-contracted curves have finite length
Stepanov, Eugene
Teplitskaya, Yana
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 2017, 96 : 455 - 481
[4] Metric and Geometric Relaxations of Self-Contracted Curves
Daniilidis, Aris
Deville, Robert
Durand-Cartagena, Estibalitz
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, 2019, 182 (01) : 81 - 109
[5] Metric and Geometric Relaxations of Self-Contracted Curves
Aris Daniilidis
Robert Deville
Estibalitz Durand-Cartagena
Journal of Optimization Theory and Applications, 2019, 182 : 81 - 109
[6] Self-contracted Curves in CAT(0)-Spaces and Their Rectifiability
Ohta, Shin-ichi
JOURNAL OF GEOMETRIC ANALYSIS, 2020, 30 (01) : 936 - 967
[7] Rectifiability of Self-Contracted Curves in the Euclidean Space and Applications
A. Daniilidis
G. David
E. Durand-Cartagena
A. Lemenant
The Journal of Geometric Analysis, 2015, 25 : 1211 - 1239
[8] Rectifiability of Self-Contracted Curves in the Euclidean Space and Applications
Daniilidis, A.
David, G.
Durand-Cartagena, E.
Lemenant, A.
JOURNAL OF GEOMETRIC ANALYSIS, 2015, 25 (02) : 1211 - 1239
[9] Self-Contracted Curves in Spaces With Weak Lower Curvature Bound
Lebedeva, Nina
Ohta, Shin-ichi
Zolotov, Vladimir
INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2021, 2021 (11) : 8623 - 8656
[10] Ubiquitous Algorithms in Convex Optimization Generate Self-Contracted Sequences
Boehm, Axel
Daniilidis, Aris
JOURNAL OF CONVEX ANALYSIS, 2022, 29 (01) : 119 - 128

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