Nothing Special   »   [go: up one dir, main page]
Skip to main content

Advertisement

Springer Nature Link
Log in

A new line search inexact restoration approach for nonlinear programming

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

A new general scheme for Inexact Restoration methods for Nonlinear Programming is introduced. After computing an inexactly restored point, the new iterate is determined in an approximate tangent affine subspace by means of a simple line search on a penalty function. This differs from previous methods, in which the tangent phase needs both a line search based on the objective function (or its Lagrangian) and a confirmation based on a penalty function or a filter decision scheme. Besides its simplicity the new scheme enjoys some nice theoretical properties. In particular, a key condition for the inexact restoration step could be weakened. To some extent this also enables the application of the new scheme to mathematical programs with complementarity constraints.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Andreani, R., Castro, S.L.C., Chela, J., Friedlander, A., Santos, S.A.: An inexact-restoration method for nonlinear bilevel programming problems. Comput. Optim. Appl. (2007). doi:10.1007/s10589-007-9147-4

    Google Scholar 

  2. Andreani, R., Martínez, J.M.: On the solution of mathematical programming problems with equilibrium constraints. Math. Methods Oper. Res. 54, 345–358 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Andreani, A., Martínez, J.M., Schuverdt, M.L.: On the relation between constant positive linear dependence condition and quasinormality constraint qualification. J. Optim. Theory Appl. 125, 473–485 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Birgin, E.G., Martínez, J.M.: Local convergence of an inexact-restoration method and numerical experiments. J. Optim. Theory Appl. 127, 229–247 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Birgin, E.G., Martínez, J.M., Raydan, M.: Algorithm 813: SPG—software for convex-constrained optimization. ACM Trans. Math. Softw. 27, 340–349 (2001)

    Article  MATH  Google Scholar 

  7. Birgin, E.G., Martinez, J.M., Raydan, M.: Inexact spectral projected gradient methods on convex sets. IMA J. Numer. Anal. 23, 539–559 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gomes-Ruggiero, M.A., Martínez, J.M., Santos, S.A.: Spectral projected gradient method with inexact restoration for minimization with nonconvex constraints. SIAM J. Sci. Comput. 31, 1628–1652 (2009)

    Article  MathSciNet  Google Scholar 

  9. Gonzaga, C.C., Karas, E.W., Vanti, M.: A globally convergent filter method for nonlinear programming. SIAM J. Optim. 14, 646–669 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Haslinger, J., Neittaanmäki, P.: Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd edn. Wiley, Chichester (1996)

    MATH  Google Scholar 

  11. Karas, E.W., Oening, A.P., Ribeiro, A.A.: Global convergence of slanting filter methods for nonlinear programming. Appl. Math. Comput. 200, 486–500 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kaya, C.Y., Martínez, J.M.: Euler discretization and Inexact restoration for optimal control. J. Optim. Theory Appl. 134, 191–206 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kaya, C.Y., Martínez, J.M.: Euler discretization and inexact restoration for optimal control. Technical Report (2007). http://www.maths.unisa.edu.au/~yalcin/techreports/iroc.pdf. Accessed 31 May 2009

  14. Martínez, J.M.: Inexact restoration method with Lagrangian tangent decrease and new merit function for nonlinear programming. J. Optim. Theory Appl. 111, 39–58 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Martínez, J.M., Pilotta, E.A.: Inexact restoration algorithms for constrained optimization. J. Optim. Theory Appl. 104, 135–163 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Martínez, J.M., Svaiter, B.F.: A practical optimality condition without constraint qualifications for nonlinear programming. J. Optim. Theory Appl. 118, 117–133 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht (1998)

    MATH  Google Scholar 

  18. Qi, L., Wei, Z.: On the constant positive linear dependence condition and its application to SQP methods. SIAM J. Optim. 10, 963–981 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Robinson, S.M.: Stability theory for systems of inequalities, part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  20. Silva, C.E.P., Monteiro, M.T.T.: A filter inexact-restoration method for nonlinear programming. TOP 16, 126–146 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Silva, C.E.P., Monteiro, M.T.T.: A filter algorithm: comparison with NLP solvers. Int. J. Comput. Math. 85, 667–689 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Numerical Mathematics, Department of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany

    Andreas Fischer

  2. Department of Applied Mathematics, IMECC-UNICAMP, University of Campinas, CP 6065, 13081-970, Campinas, SP, Brazil

    Ana Friedlander

Authors
  1. Andreas Fischer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ana Friedlander
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Fischer.

Additional information

This work was supported by PRONEX-Optimization (PRONEX - CNPq / FAPERJ E-26 / 171.510/2006 - APQ1), FAPESP (Grants 2006/53768-0 and 2008/00875-9) and CNPq.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fischer, A., Friedlander, A. A new line search inexact restoration approach for nonlinear programming. Comput Optim Appl 46, 333–346 (2010). https://doi.org/10.1007/s10589-009-9267-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-009-9267-0

Keywords

Search

Navigation