Mathematical Programming

, Volume 100, Issue 2, pp 379–410 | Cite as

A globally convergent primal-dual interior-point filter method for nonlinear programming

  • Michael Ulbrich
  • Stefan Ulbrich
  • Luís N. Vicente
Article

Abstract.

In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primal-dual interior-point algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters.

The new algorithm decomposes the primal-dual step obtained from the perturbed first-order necessary conditions into a normal and a tangential step, whose sizes are controlled by a trust-region type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step.

Global convergence to first-order critical points is proved for the new primal-dual interior-point filter algorithm.

Keywords

interior-point methods primal-dual filter global convergence 

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References

  1. 1.
    Argaez, M., Tapia, R.A.: On the global convergence of a modified augmented Lagrangian line search interior point Newton method for nonlinear programming. Tech. Report TR95–38, Department of Computational and Applied Mathematics, Rice University, 1995. Revised September 1999Google Scholar
  2. 2.
    Audet, C., Dennis, J.E.: A pattern search filter method for nonlinear programming without derivatives. Tech. Report TR00–09, Department of Computational and Applied Mathematics, Rice University, 2000Google Scholar
  3. 3.
    Benson, H.Y., Shanno, D.F., Vanderbei, R.J.: Interior-point methods for nonconvex nonlinear programming: Filter methods and merit functions. Tech. Report ORFE-00-06, Operations Research and Financial Engineering, Princeton University, 2000Google Scholar
  4. 4.
    Boggs, P.T., Tolle, J.W.: Sequential quadratic programming. In: Acta Numerica 1995, A. Iserles, (ed.), Cambridge University Press, Cambridge, London, New York, 1995, pp. 1–51Google Scholar
  5. 5.
    Byrd, R.H., Hribar, M.E., Nocedal, J.: An interior point algorithm for large-scale nonlinear programming. SIAM J. Optim. 9, 877–900 (1999)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Byrd, R.H., Liu, G., Nocedal, J.: On the local behavior of an interior point method for nonlinear programming. In: Numerical analysis 1997 (Dundee) Pitman Res. Notes Math. Ser., 380, Longman, Harlow, 1998, pp. 37–56Google Scholar
  7. 7.
    Chin, C.M., Fletcher, R.: On the global convergence of an SLP-filter algorithm that takes EQP steps. Tech. Report NA/199, Department of Mathematics, University of Dundee, 2001Google Scholar
  8. 8.
    Coleman, T.F., Li, Y.: On the convergence of interior–reflective Newton methods for nonlinear minimization subject to bounds. Math. Program. 67, 189–224 (1994)MathSciNetGoogle Scholar
  9. 9.
    Conn, A.R., Gould, N.I.M., Orban, D., Toint P.L.: A primal-dual trust-region algorithm for minimizing a non-convex function subject to general inequality and linear equality constraints. In: Nonlinear optimization and related topics (Erice, 1998), Kluwer Acad. Publ., Dordrecht, 2000, pp. 15–49Google Scholar
  10. 10.
    Conn, A.R., Gould, N.I.M., Toint P.L.: Trust-Region Methods. MPS-SIAM Series on Optimization, SIAM, Philadelphia, 2000Google Scholar
  11. 11.
    El-Bakry, A.S., Tapia, R.A., Tsuchiya, T., Zhang, Y.: On the formulation and theory of the Newton interior–point method for nonlinear programming. J. Optim. Theor. Appl. 89, 507–541 (1996)MathSciNetMATHGoogle Scholar
  12. 12.
    Fletcher, R., Gould, N.I.M., Leyffer, S., Toint, P.L., Wächter, A.: Global convergence of trust-region SQP-filter algorithms for general nonlinear programming. SIAM J. Optim. 13, 635–659 (2002)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Fletcher, R., Leyffer, S.: User manual for filterSQP. Tech. Report NA/181, Department of Mathematics, University of Dundee, 1998Google Scholar
  14. 14.
    Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Math. Programm. 91, 239–269 (2002)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Fletcher, R., Leyffer, S., Toint, P.L.: On the global convergence of an SLP-filter algorithm. Tech. Report 98/13, Département de Mathématique, FUNDP, Namur, 1998Google Scholar
  16. 16.
    Forsgren, A., Gill, P.E.: Primal-dual interior methods for nonconvex nonlinear programming. SIAM J. Optim. 8, 1132–1152 (1998)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Gay, D.M., Overton, M.L., Wright, M.H.: A primal-dual interior method for nonconvex nonlinear programming. In: Proceedings of the 1996 International Conference on Nonlinear Programming, Beijing, China, Y. Yuan, (ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998, pp. 31–56Google Scholar
  18. 18.
    Heinkenschloss, M., Ulbrich, M., Ulbrich, S.: Superlinear and quadratic convergence of affine-scaling interior-point Newton methods for problems with simple bounds without strict complementarity assumption. Math. Program. 86, 615–635 (1999)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Martinez, H.J., Parada, Z., Tapia, R.A.: On the characterization of q-superlinear convergence of quasi-Newton interior-point methods for nonlinear programming. Boletin de la Sociedad Matematica Mexicana 1, 1–12 (1995)Google Scholar
  20. 20.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer–Verlag, Berlin, 1999Google Scholar
  21. 21.
    Tseng,P.: Error bounds and superlinear convergence analysis of some Newton-type methods in optimization. In: Nonlinear Optimization and Applications, Vol. 2, Kluwer Academic Publishers B.V., 1998Google Scholar
  22. 22.
    Ulbrich, M., Ulbrich, S.: Nonmonotone trust region methods for nonlinear equality constrained optimization without a penalty function. Math. Program. 95, 103–135 (2003)CrossRefGoogle Scholar
  23. 23.
    Vanderbei, R.J., Shanno, D.F.: An interior-point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13, 231–252 (1999)Google Scholar
  24. 24.
    Vicente, L.N.: Local convergence of the affine-scaling interior-point algorithm for nonlinear programming. Comput. Optim. Appl. 17, 23–35 (2000)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Vicente, L.N., Wright, S.J.: Local convergence of a primal-dual method for degenerate nonlinear programming. Comput. Optim. Appl. 22, 311–328 (2002)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Wächter, A.: An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering. PhD thesis, Department of Chemical Engineering, Carnegie Mellon University, 2002Google Scholar
  27. 27.
    Wächter, W., Biegler, L.T.: Failure of global convergence for a class of interior point methods for nonlinear programming. Math. Program. 88, (2000)Google Scholar
  28. 28.
    Wächter, A., Biegler, L.T.: Global and local convergence of line search filter methods for nonlinear programming. Tech. Report B-01-09, CAPD, Department of Chemical Engineering, Carnegie Mellon University, 2001Google Scholar
  29. 29.
    Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia, 1997Google Scholar
  30. 30.
    Wright, S.J., Orban, D.: Properties of the log-barrier function on degenerate nonlinear programs. Math. Oper. Res. 27, 585–613 (2003)CrossRefGoogle Scholar
  31. 31.
    Yamashita, H., Yabe, H.: Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization. Math. Program. 75, 377–397 (1996)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Michael Ulbrich
    • 1
  • Stefan Ulbrich
    • 2
  • Luís N. Vicente
    • 3
  1. 1.Schwerpunkt Optimierung und ApproximationFachbereich MathematikHamburgGermany
  2. 2.Zentrum Mathematik M1Technische Universität MünchenGarching b. MünchenGermany
  3. 3.Departamento de MatemáticaUniversidade de CoimbraCoimbraPortugal

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