Approaching nonsmooth nonconvex minimization through second order proximal-gradient dynamical systems

Radu Ioan Bot; Ernö Robert Csetnek; Szilard Laszlo

doi:10.1007/s00028-018-0441-7

Approaching nonsmooth nonconvex minimization through second order proximal-gradient dynamical systems

Radu Ioan Bot (Corresponding author), Ernö Robert Csetnek, Szilard Laszlo

Department of Mathematics

Publications: Contribution to journal › Article › Peer Reviewed

Abstract

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection with the minimization of the sum of a nonsmooth convex and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka–Łojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Łojasiewicz exponent.

Original language	English
Pages (from-to)	1291-1318
Number of pages	28
Journal	Journal of Evolution Equations
Volume	18
Issue number	3
DOIs	https://doi.org/10.1007/s00028-018-0441-7
Publication status	Published - Sept 2018

Austrian Fields of Science 2012

101002 Analysis
101027 Dynamical systems

Keywords

ALGORITHMS
CONVERGENCE
INCLUSIONS
Kurdyka-ojasiewicz property
Limiting subdifferential
MAXIMAL MONOTONE-OPERATORS
Nonsmooth nonconvex optimization
Second-order dynamical system
Kurdyka–Łojasiewicz property

Access to Document

10.1007/s00028-018-0441-7Licence: CC BY 4.0

http://phaidra.univie.ac.at/o:937310Licence: CC BY 4.0

Cite this

@article{5009009c06714869bb97e7a2931a6341,

title = "Approaching nonsmooth nonconvex minimization through second order proximal-gradient dynamical systems",

abstract = "We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection with the minimization of the sum of a nonsmooth convex and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka–{\L}ojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the {\L}ojasiewicz exponent.",

keywords = "ALGORITHMS, CONVERGENCE, INCLUSIONS, Kurdyka-ojasiewicz property, Limiting subdifferential, MAXIMAL MONOTONE-OPERATORS, Nonsmooth nonconvex optimization, Second-order dynamical system, Kurdyka–{\L}ojasiewicz property",

author = "Bot, {Radu Ioan} and Csetnek, {Ern{\"o} Robert} and Szilard Laszlo",

note = "Publisher Copyright: {\textcopyright} 2018, The Author(s).",

year = "2018",

month = sep,

doi = "10.1007/s00028-018-0441-7",

language = "English",

volume = "18",

pages = "1291--1318",

journal = "Journal of Evolution Equations",

issn = "1424-3199",

number = "3",

}

TY - JOUR

T1 - Approaching nonsmooth nonconvex minimization through second order proximal-gradient dynamical systems

AU - Bot, Radu Ioan

AU - Csetnek, Ernö Robert

AU - Laszlo, Szilard

PY - 2018/9

Y1 - 2018/9

N2 - We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection with the minimization of the sum of a nonsmooth convex and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka–Łojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Łojasiewicz exponent.

AB - We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection with the minimization of the sum of a nonsmooth convex and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka–Łojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Łojasiewicz exponent.

KW - ALGORITHMS

KW - CONVERGENCE

KW - INCLUSIONS

KW - Kurdyka-ojasiewicz property

KW - Limiting subdifferential

KW - MAXIMAL MONOTONE-OPERATORS

KW - Nonsmooth nonconvex optimization

KW - Second-order dynamical system

KW - Kurdyka–Łojasiewicz property

UR - http://www.scopus.com/inward/record.url?scp=85042620850&partnerID=8YFLogxK

U2 - 10.1007/s00028-018-0441-7

DO - 10.1007/s00028-018-0441-7

M3 - Article

SN - 1424-3199

VL - 18

SP - 1291

EP - 1318

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

IS - 3

ER -

Approaching nonsmooth nonconvex minimization through second order proximal-gradient dynamical systems

Abstract

Austrian Fields of Science 2012

Keywords

Access to Document

Other files and links

Fingerprint

Cite this