Abstract
In this paper, we deal with matrix-free preconditioners for nonlinear conjugate gradient (NCG) methods. In particular, we review proposals based on quasi-Newton updates, and either satisfying the secant equation or a secant-like equation at some of the previous iterates. Conditions are given proving that, in some sense, the proposed preconditioners also approximate the inverse of the Hessian matrix. In particular, the structure of the preconditioners depends both on low-rank updates along with some specific parameters. The low-rank updates are obtained as by-product of NCG iterations. Moreover, we consider the possibility to embed damped techniques within a class of preconditioners based on quasi-Newton updates. Damped methods have proved to be effective to enhance the performance of quasi-Newton updates, in those cases where the Wolfe linesearch conditions are hardly fulfilled. The purpose is to extend the idea behind damped methods also to improve NCG schemes, following a novel line of research in the literature. The results, which summarize an extended numerical experience using large-scale CUTEst problems, is reported, showing that these approaches can considerably improve the performance of NCG methods.
| Original language | English |
|---|---|
| Title of host publication | Numerical Analysis and Optimization - NAO-IV, 2017 |
| Editors | Lucio Grandinetti, Mehiddin Al-Baali, Anton Purnama |
| Publisher | Springer New York LLC |
| Pages | 1-21 |
| Number of pages | 21 |
| Volume | 235 |
| ISBN (Print) | 9783319900254 |
| DOIs | |
| Publication status | Published - Jan 1 2018 |
| Event | 4th International Conference on Numerical Analysis and Optimization, NAO-IV 2017 - Muscat, Oman Duration: Jan 2 2017 → Jan 5 2017 |
Other
| Other | 4th International Conference on Numerical Analysis and Optimization, NAO-IV 2017 |
|---|---|
| Country | Oman |
| City | Muscat |
| Period | 1/2/17 → 1/5/17 |
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Keywords
- Conjugate gradient method
- Damped techniques
- Large scale unconstrained optimization
- Nonlinear conjugate gradient methods
- Preconditioning
- Quasi-Newton methods
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Quasi-newton based preconditioning and damped quasi-newton schemes for nonlinear conjugate gradient methods. / Al-Baali, Mehiddin; Caliciotti, Andrea; Fasano, Giovanni; Roma, Massimo.
Numerical Analysis and Optimization - NAO-IV, 2017. ed. / Lucio Grandinetti; Mehiddin Al-Baali; Anton Purnama. Vol. 235 Springer New York LLC, 2018. p. 1-21.Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
TY - GEN
T1 - Quasi-newton based preconditioning and damped quasi-newton schemes for nonlinear conjugate gradient methods
AU - Al-Baali, Mehiddin
AU - Caliciotti, Andrea
AU - Fasano, Giovanni
AU - Roma, Massimo
PY - 2018/1/1
Y1 - 2018/1/1
N2 - In this paper, we deal with matrix-free preconditioners for nonlinear conjugate gradient (NCG) methods. In particular, we review proposals based on quasi-Newton updates, and either satisfying the secant equation or a secant-like equation at some of the previous iterates. Conditions are given proving that, in some sense, the proposed preconditioners also approximate the inverse of the Hessian matrix. In particular, the structure of the preconditioners depends both on low-rank updates along with some specific parameters. The low-rank updates are obtained as by-product of NCG iterations. Moreover, we consider the possibility to embed damped techniques within a class of preconditioners based on quasi-Newton updates. Damped methods have proved to be effective to enhance the performance of quasi-Newton updates, in those cases where the Wolfe linesearch conditions are hardly fulfilled. The purpose is to extend the idea behind damped methods also to improve NCG schemes, following a novel line of research in the literature. The results, which summarize an extended numerical experience using large-scale CUTEst problems, is reported, showing that these approaches can considerably improve the performance of NCG methods.
AB - In this paper, we deal with matrix-free preconditioners for nonlinear conjugate gradient (NCG) methods. In particular, we review proposals based on quasi-Newton updates, and either satisfying the secant equation or a secant-like equation at some of the previous iterates. Conditions are given proving that, in some sense, the proposed preconditioners also approximate the inverse of the Hessian matrix. In particular, the structure of the preconditioners depends both on low-rank updates along with some specific parameters. The low-rank updates are obtained as by-product of NCG iterations. Moreover, we consider the possibility to embed damped techniques within a class of preconditioners based on quasi-Newton updates. Damped methods have proved to be effective to enhance the performance of quasi-Newton updates, in those cases where the Wolfe linesearch conditions are hardly fulfilled. The purpose is to extend the idea behind damped methods also to improve NCG schemes, following a novel line of research in the literature. The results, which summarize an extended numerical experience using large-scale CUTEst problems, is reported, showing that these approaches can considerably improve the performance of NCG methods.
KW - Conjugate gradient method
KW - Damped techniques
KW - Large scale unconstrained optimization
KW - Nonlinear conjugate gradient methods
KW - Preconditioning
KW - Quasi-Newton methods
UR - http://www.scopus.com/inward/record.url?scp=85048243750&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85048243750&partnerID=8YFLogxK
U2 - 10.1007/978-3-319-90026-1_1
DO - 10.1007/978-3-319-90026-1_1
M3 - Conference contribution
AN - SCOPUS:85048243750
SN - 9783319900254
VL - 235
SP - 1
EP - 21
BT - Numerical Analysis and Optimization - NAO-IV, 2017
A2 - Grandinetti, Lucio
A2 - Al-Baali, Mehiddin
A2 - Purnama, Anton
PB - Springer New York LLC
ER -