A positive barzilai-borwein-like stepsize and an extension for symmetric linear systems

Yu Hong Dai; Mehiddin Al-Baali; Xiaoqi Yang

doi:10.1007/978-3-319-17689-5_3

A positive barzilai-borwein-like stepsize and an extension for symmetric linear systems

Yu Hong Dai^*, Mehiddin Al-Baali, Xiaoqi Yang

^*Corresponding author for this work

Mathematics & Statistics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

22 Citations (Scopus)

Abstract

The Barzilai and Borwein (BB) gradient method has achieved a lot of attention since it performs much more better than the classical steepest descent method. In this paper, we analyze a positive BB-like gradient stepsize and discuss its possible uses. Specifically, we present an analysis of the positive stepsize for two-dimensional strictly convex quadratic functions and prove the R-superlinear convergence under some assumption. Meanwhile, we extend BB-like methods for solving symmetric linear systems and find that a variant of the positive stepsize is very useful in the context. Some useful discussions on the positive stepsize are also given.

Original language	English
Title of host publication	Springer Proceedings in Mathematics and Statistics
Publisher	Springer New York LLC
Pages	59-75
Number of pages	17
Volume	134
ISBN (Print)	9783319176888
DOIs	https://doi.org/10.1007/978-3-319-17689-5_3
Publication status	Published - 2015
Event	3rd International Conference on Numerical Analysis and Optimization: Theory, Methods, Applications and Technology Transfer, NAOIII-2014 - Muscat, Oman Duration: Jan 5 2014 → Jan 9 2014

Other

Other	3rd International Conference on Numerical Analysis and Optimization: Theory, Methods, Applications and Technology Transfer, NAOIII-2014
Country/Territory	Oman
City	Muscat
Period	1/5/14 → 1/9/14

Keywords

Barzilai and Borwein gradient method
Condition number
Quadratic function
R-superlinear convergence
Unconstrained optimization

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.1007/978-3-319-17689-5_3

Cite this

Dai, YH, Al-Baali, M & Yang, X 2015, A positive barzilai-borwein-like stepsize and an extension for symmetric linear systems. in Springer Proceedings in Mathematics and Statistics. vol. 134, Springer New York LLC, pp. 59-75, 3rd International Conference on Numerical Analysis and Optimization: Theory, Methods, Applications and Technology Transfer, NAOIII-2014, Muscat, Oman, 1/5/14. https://doi.org/10.1007/978-3-319-17689-5_3

@inproceedings{b29bb67064ec44caaa68e183c78a637b,

title = "A positive barzilai-borwein-like stepsize and an extension for symmetric linear systems",

abstract = "The Barzilai and Borwein (BB) gradient method has achieved a lot of attention since it performs much more better than the classical steepest descent method. In this paper, we analyze a positive BB-like gradient stepsize and discuss its possible uses. Specifically, we present an analysis of the positive stepsize for two-dimensional strictly convex quadratic functions and prove the R-superlinear convergence under some assumption. Meanwhile, we extend BB-like methods for solving symmetric linear systems and find that a variant of the positive stepsize is very useful in the context. Some useful discussions on the positive stepsize are also given.",

keywords = "Barzilai and Borwein gradient method, Condition number, Quadratic function, R-superlinear convergence, Unconstrained optimization",

author = "Dai, {Yu Hong} and Mehiddin Al-Baali and Xiaoqi Yang",

year = "2015",

doi = "10.1007/978-3-319-17689-5_3",

language = "English",

isbn = "9783319176888",

volume = "134",

pages = "59--75",

booktitle = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer New York LLC",

note = "3rd International Conference on Numerical Analysis and Optimization: Theory, Methods, Applications and Technology Transfer, NAOIII-2014 ; Conference date: 05-01-2014 Through 09-01-2014",

}

TY - GEN

T1 - A positive barzilai-borwein-like stepsize and an extension for symmetric linear systems

AU - Dai, Yu Hong

AU - Al-Baali, Mehiddin

AU - Yang, Xiaoqi

PY - 2015

Y1 - 2015

N2 - The Barzilai and Borwein (BB) gradient method has achieved a lot of attention since it performs much more better than the classical steepest descent method. In this paper, we analyze a positive BB-like gradient stepsize and discuss its possible uses. Specifically, we present an analysis of the positive stepsize for two-dimensional strictly convex quadratic functions and prove the R-superlinear convergence under some assumption. Meanwhile, we extend BB-like methods for solving symmetric linear systems and find that a variant of the positive stepsize is very useful in the context. Some useful discussions on the positive stepsize are also given.

AB - The Barzilai and Borwein (BB) gradient method has achieved a lot of attention since it performs much more better than the classical steepest descent method. In this paper, we analyze a positive BB-like gradient stepsize and discuss its possible uses. Specifically, we present an analysis of the positive stepsize for two-dimensional strictly convex quadratic functions and prove the R-superlinear convergence under some assumption. Meanwhile, we extend BB-like methods for solving symmetric linear systems and find that a variant of the positive stepsize is very useful in the context. Some useful discussions on the positive stepsize are also given.

KW - Barzilai and Borwein gradient method

KW - Condition number

KW - Quadratic function

KW - R-superlinear convergence

KW - Unconstrained optimization

UR - http://www.scopus.com/inward/record.url?scp=84947050729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947050729&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-17689-5_3

DO - 10.1007/978-3-319-17689-5_3

M3 - Conference contribution

AN - SCOPUS:84947050729

SN - 9783319176888

VL - 134

SP - 59

EP - 75

BT - Springer Proceedings in Mathematics and Statistics

PB - Springer New York LLC

T2 - 3rd International Conference on Numerical Analysis and Optimization: Theory, Methods, Applications and Technology Transfer, NAOIII-2014

Y2 - 5 January 2014 through 9 January 2014

ER -

A positive barzilai-borwein-like stepsize and an extension for symmetric linear systems

Abstract

Other

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this