Advertisement

On Minimizing Objective and KKT Error in a Filter Line Search Strategy for an Interior Point Method

  • M. Fernanda P. Costa
  • Edite M. G. P. Fernandes
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6784)

Abstract

This paper carries out a numerical study of filter line search strategies that aim at minimizing the objective function and the Karush-Kuhn-Tucker (KKT) vector error in order to encourage global convergence of interior point methods. These filter strategies are implemented in an infeasible primal-dual interior point framework for nonlinear programming. First, we propose a filter that has four components measuring primal feasibility, complementarity, dual feasibility and optimality. The different measures arise from the KKT conditions of the problem. Then, we combine the KKT equations defining a different two-dimensional filter technique. The versions have in common the objective function as the optimality measure. The primary assessment of these techniques has been done with a well-known collection of small- and medium-scale problems.

Keywords

Nonlinear programming Interior point method Filter method Line search 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benson, H.Y., Vanderbei, R.J., Shanno, D.F.: Interior-point methods for nonconvex nonlinear programming:filter methods and merit functions. Computational Optimization and Applications 23, 257–272 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Costa, M.F.P., Fernandes, E.M.G.P.: Comparison of interior point filter line search strategies for constrained optimization by performance profiles. International Journal of Mathematics Models and Methods in Applied Sciences 1, 111–116 (2007)Google Scholar
  3. 3.
    Costa, M.F.P., Fernandes, E.M.G.P.: Practical implementation of an interior point nonmonotone line search filter method. International Journal of Computer Mathematics 85, 397–409 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Mathematical Programming 91, 201–213 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Mathematical Programming 91, 239–269 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Fletcher, R., Leyffer, S., Toint, P.: A brief history of filter methods, Report ANL/MCS-P1372-0906, Argonne National Laboratory (2006)Google Scholar
  7. 7.
    Fourer, R., Gay, D.M., Kernighan, B.: A modeling language for mathematical programming. Management Science 36, 519–554 (1990)CrossRefzbMATHGoogle Scholar
  8. 8.
    Gould, N.I.M., Orban, D., Sartenaer, A., Toint, P.L.: Superlinear convergence of primal-dual interior point algorithms for nonlinear programming. SIAM Journal on Optimization 11, 974–1002 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Shanno, D.F., Vanderbei, R.J.: Interior-point methods for nonconvex nonlinear programming: orderings and higher-order methods. Mathematical Programming B 87, 303–316 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Ulbrich, M., Ulbrich, S., Vicente, L.N.: A globally convergent primal-dual interior-point filter method for nonlinear programming. Mathematical Programming 100, 379–410 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Vanderbei, R.J.: LOQO: An interior-code for quadratic programming. Technical report SOR-94-15, Princeton University, Statistics and Operations Research (1998)Google Scholar
  12. 12.
    Vanderbei, R.J., Shanno, D.F.: An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications 13, 231–252 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM Journal on Optimization 16, 1–31 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: local convergence. SIAM Journal on Optimization 16, 32–48 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming 106, 25–57 (2007)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • M. Fernanda P. Costa
    • 1
  • Edite M. G. P. Fernandes
    • 2
  1. 1.Department of Mathematics and ApplicationsUniversity of MinhoPortugal
  2. 2.Algoritmi R&D CentreUniversity of MinhoBragaPortugal

Personalised recommendations