Skip to main content
SpringerLink
Log in

Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization, Part 2: Application

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we describe an application of the planar conjugate gradient method introduced in Part 1 (Ref. 1) and aimed at solving indefinite nonsingular sets of linear equations. We prove that it can be used fruitfully within optimization frameworks; in particular, we present a globally convergent truncated Newton scheme, which uses the above planar method for solving the Newton equation. Finally, our approach is tested over several problems from the CUTE collection (Ref. 2).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Fasano (2005) ArticleTitle Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization, Part 1: Theory Journal of Optimization Theory and Applications 125 523–541

    Google Scholar 

  2. I. Bongartz A. R. Conn N. Gould P. L. Toint (1995) ArticleTitle CUTE: Constrained and Unconstrained Test Environment ACM Transactions on Mathematical Software 21 123–160 Occurrence Handle10.1145/200979.201043

    Article  Google Scholar 

  3. O. Axellson (1996) Iterative Solution Methods Cambridge University Press Cambridge UK

    Google Scholar 

  4. D. P. Bertsekas (1995) Nonlinear Programming Athena Scientific Belmont Massachusetts

    Google Scholar 

  5. G. P. McCormick (1983) Nonlinear Programming: Theory, Algorithm, and Applications Wiley and Sons New York, NY

    Google Scholar 

  6. L. Grippo F. Lampariello S. Lucidi (1989) ArticleTitle A Truncated Newton Method with Nonmonotone Line Search for Unconstrained Optimization Journal of Optimization Theory and Applications 60 401–419 Occurrence Handle10.1007/BF00940345

    Article  Google Scholar 

  7. R. S. Dembo T. Steihaug (1983) ArticleTitle Truncated Newton Algorithms for Large-Scale Matrix Methods Mathematical Programming 26 190–212

    Google Scholar 

  8. L. Grippo F. Lampariello S. Lucidi (1986) ArticleTitle A Nonmonotone Line Search Technique for Newton’s Method SIAM Journal on Numerical Analysis 23 707–716 Occurrence Handle10.1137/0723046

    Article  Google Scholar 

  9. R. S. Dembo S. C. Eisenstat T. Steihaug (1982) ArticleTitle Inexact Newton Methods SIAM Journal on Numerical Analysis 19 400–408 Occurrence Handle10.1137/0719025

    Article  Google Scholar 

  10. S. G. Nash (2000) ArticleTitle A Survey of Truncated Newton Methods Journal of Computational and Applied Mathematics 124 45–59 Occurrence Handle10.1016/S0377-0427(00)00426-X Occurrence HandleMR1803293

    Article  MathSciNet  Google Scholar 

  11. S. Lucidi M. Roma (1997) ArticleTitle Numerical Experiences with Truncated Newton Methods in Large-Scale Unconstrained Optimization Computational Optimization and Applications 7 71–87 Occurrence Handle10.1023/A:1008619812615

    Article  Google Scholar 

  12. J. J. More’ D. C. Sorensen (1979) ArticleTitle On the Use of Directions of Negative Curvature in a Modified Newton Method Mathematical Programming 16 1–20 Occurrence Handle10.1007/BF01582091

    Article  Google Scholar 

  13. S. Lucidi F. Rochetich M. Roma (1999) ArticleTitle Curvilinear Stabilization Techniques for Truncated Newton Method in Large-Scale Unconstrained Optimization SIAM Journal on Optimization 8 916–939 Occurrence Handle10.1137/S1052623495295250

    Article  Google Scholar 

  14. N. I. M. Gould S. Lucidi M. Roma P. L. Toint (2000) ArticleTitle Exploiting Negative Curvature Directions in Line Search Methods for Unconstrained Optimization Optimization Methods and Software 14 75–98

    Google Scholar 

  15. S. C. Eisenstat H. F. Walker (1994) Choosing the Forcing Terms in an Inexact Newton Method Rice University Houston Texas

    Google Scholar 

  16. M. R. Hestenes (1980) Conjugate Direction Methods in Optimization Springer Verlag New York, NY

    Google Scholar 

  17. D. G. Luenberger (1969) ArticleTitle Hyperbolic Pairs in the Method of Conjugate Gradients SIAM Journal on Applied Mathematics 17 1263–1267

    Google Scholar 

  18. G. Fasano (2001) Use of Conjugate Directions Inside Newton-Type Algorithms for Large-Scale Unconstrained Optimization Italy Rome

    Google Scholar 

  19. M. Srinivasan (1994) Using Directions of Negative Curvature in Newton-Type Methods for Nonlinear Nonconvex Problems School of Information Technology and Engineering, George Mason University Fairfax Virginia

    Google Scholar 

  20. J. E. Dennis R. B. Schnabel (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations Prentice-Hall, Englewood Cliffs New Jersey

    Google Scholar 

  21. C. C. Paige M. A. Saunders (1975) ArticleTitleSolution of Sparse Indefinite Systems of Linear Equations SIAM Journal on Numerical Analysis 12 617–629 Occurrence Handle10.1137/0712047

    Article  Google Scholar 

  22. G. Fasano (2004) Planar Conjugate-Gradient Algorithm for Large-Scale Unconstrained Optimization Part 2: Applications Istituto Nazionale per Studi ed Esperienze di Architettura Navale (INSEAN) Rome Italy

    Google Scholar 

  23. L. Grippo F. Lampariello S. Lucidi (1991) ArticleTitleA Class of Nonmonotone Stabilization Methods in Unconstrained Optimization Numerische Mathematik 59 779–805 Occurrence Handle10.1007/BF01385810

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Postdoctoral Fellow, Department of Computer Science and Systems A. Ruberti, University of Rome-La Sapienza, Rome, Italy

    G. Fasano

Authors
  1. G. Fasano
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by MIUR, FIRB Research Program on Large-Scale Nonlinear Optimization, Rome, Italy.

The author acknowledges Luigi Grippo and Stefano Lucidi, who contributed considerably to the elaboration of this paper. The exchange of experiences with Massimo Roma was a constant help in the investigation. The author expresses his gratitude to the Associate Editor and the referees for suggestions and corrections.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fasano, G. Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization, Part 2: Application. J Optim Theory Appl 125, 543–558 (2005). https://doi.org/10.1007/s10957-005-2088-0

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-005-2088-0

Keywords

Search

Navigation