On Nesterov's nonsmooth ChebyshevRosenbrock functions

Mert Grbzbalaban, Michael L. Overton

Research output: Contribution to journalArticle

Abstract

We discuss two nonsmooth functions on R n introduced by Nesterov. We show that the first variant is partly smooth in the sense of Lewis and that its only stationary point is the global minimizer. In contrast, we show that the second variant has 2 n-1 Clarke stationary points, none of them local minimizers except the global minimizer, but also that its only Mordukhovich stationary point is the global minimizer. Nonsmooth optimization algorithms from multiple starting points generate iterates that approximate all 2 n-1 Clarke stationary points, not only the global minimizer, but it remains an open question as to whether the nonminimizing Clarke stationary points are actually points of attraction for optimization algorithms.

Original languageEnglish (US)
Pages (from-to)1282-1289
Number of pages8
JournalNonlinear Analysis, Theory, Methods and Applications
Volume75
Issue number3
DOIs
StatePublished - Feb 2012

Fingerprint

Nonsmooth Function
Stationary point
Global Minimizer
Optimization Algorithm
Nonsmooth Optimization
Local Minimizer
Iterate

Keywords

  • Nonsmooth optimization
  • Optimization algorithms
  • Variational analysis

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

On Nesterov's nonsmooth ChebyshevRosenbrock functions. / Grbzbalaban, Mert; Overton, Michael L.

In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 75, No. 3, 02.2012, p. 1282-1289.

Research output: Contribution to journalArticle

@article{ed87560e61f34eaa94ce255caf38f659,
title = "On Nesterov's nonsmooth ChebyshevRosenbrock functions",
abstract = "We discuss two nonsmooth functions on R n introduced by Nesterov. We show that the first variant is partly smooth in the sense of Lewis and that its only stationary point is the global minimizer. In contrast, we show that the second variant has 2 n-1 Clarke stationary points, none of them local minimizers except the global minimizer, but also that its only Mordukhovich stationary point is the global minimizer. Nonsmooth optimization algorithms from multiple starting points generate iterates that approximate all 2 n-1 Clarke stationary points, not only the global minimizer, but it remains an open question as to whether the nonminimizing Clarke stationary points are actually points of attraction for optimization algorithms.",
keywords = "Nonsmooth optimization, Optimization algorithms, Variational analysis",
author = "Mert Grbzbalaban and Overton, {Michael L.}",
year = "2012",
month = "2",
doi = "10.1016/j.na.2011.07.062",
language = "English (US)",
volume = "75",
pages = "1282--1289",
journal = "Nonlinear Analysis, Theory, Methods and Applications",
issn = "0362-546X",
publisher = "Elsevier Limited",
number = "3",

}

TY - JOUR

T1 - On Nesterov's nonsmooth ChebyshevRosenbrock functions

AU - Grbzbalaban, Mert

AU - Overton, Michael L.

PY - 2012/2

Y1 - 2012/2

N2 - We discuss two nonsmooth functions on R n introduced by Nesterov. We show that the first variant is partly smooth in the sense of Lewis and that its only stationary point is the global minimizer. In contrast, we show that the second variant has 2 n-1 Clarke stationary points, none of them local minimizers except the global minimizer, but also that its only Mordukhovich stationary point is the global minimizer. Nonsmooth optimization algorithms from multiple starting points generate iterates that approximate all 2 n-1 Clarke stationary points, not only the global minimizer, but it remains an open question as to whether the nonminimizing Clarke stationary points are actually points of attraction for optimization algorithms.

AB - We discuss two nonsmooth functions on R n introduced by Nesterov. We show that the first variant is partly smooth in the sense of Lewis and that its only stationary point is the global minimizer. In contrast, we show that the second variant has 2 n-1 Clarke stationary points, none of them local minimizers except the global minimizer, but also that its only Mordukhovich stationary point is the global minimizer. Nonsmooth optimization algorithms from multiple starting points generate iterates that approximate all 2 n-1 Clarke stationary points, not only the global minimizer, but it remains an open question as to whether the nonminimizing Clarke stationary points are actually points of attraction for optimization algorithms.

KW - Nonsmooth optimization

KW - Optimization algorithms

KW - Variational analysis

UR - http://www.scopus.com/inward/record.url?scp=82155173479&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=82155173479&partnerID=8YFLogxK

U2 - 10.1016/j.na.2011.07.062

DO - 10.1016/j.na.2011.07.062

M3 - Article

AN - SCOPUS:82155173479

VL - 75

SP - 1282

EP - 1289

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 3

ER -