A filter trust-region algorithm for unconstrained optimization with strong global convergence properties

Abstract

We present a new filter trust-region approach for solving unconstrained nonlinear optimization problems making use of the filter technique introduced by Fletcher and Leyffer to generate non-monotone iterations. We also use the concept of a multidimensional filter used by Gould et al. (SIAM J. Optim. 15(1):17–38, 2004) and introduce a new filter criterion showing good properties. Moreover, we introduce a new technique for reducing the size of the filter. For the algorithm, we present two different convergence analyses. First, we show that at least one of the limit points of the sequence of the iterates is first-order critical. Second, we prove the stronger property that all the limit points are first-order critical for a modified version of our algorithm. We also show that, under suitable conditions, all the limit points are second-order critical. Finally, we compare our algorithm with a natural trust-region algorithm and the filter trust-region algorithm of Gould et al. on the CUTEr unconstrained test problems Gould et al. in ACM Trans. Math. Softw. 29(4):373–394, 2003. Numerical results demonstrate the efficiency and robustness of our proposed algorithms.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Bonnans, J.F., Panier, E.R., Tits, A.L., Zhou, J.L.: Avoiding the Maratos effect by means of a nonmonotone line search. II. Inequality constrained problems feasible iterates. SIAM J. Numer. Anal. 29(4), 1187–1202 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Chin, C.M., Fletcher, R.: On the global convergence of an SLP-filter algorithm that takes EQP steps. Math. Program., Ser. A 96(1), 161–177 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods, MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000)

    Google Scholar 

  4. 4.

    Dai, Y.H.: On the nonmonotone line search. J. Optim. Theory Appl. 112(2), 315–330 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Deng, N.Y., Xiao, Y., Zhou, F.J.: Nonmonotonic trust region algorithm. J. Optim. Theory Appl. 76(2), 259–285 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program., Ser. A 91(2), 201–213 (2002)

    MATH  Article  Google Scholar 

  7. 7.

    Fletcher, R., Gould, N.I.M., Leyffer, S., Toint, P.L., Wächter, A.: Global convergence of a trust-region SQP-filter algorithm for general nonlinear programming. SIAM J. Optim. 13(3), 635–659 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Math. Program., Ser. A 91(2), 239–269 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Gould, N.I.M., Leyffer, S., Toint, P.L.: A multidimensional filter algorithm for nonlinear equations and nonlinear least-squares. SIAM J. Optim. 15(1), 17–38 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Gould, N.I.M., Lucidi, S., Roma, M., Toint, P.L.: Solving the trust-region subproblem using the Lanczos method. SIAM J. Optim. 9(2), 504–525 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Gould, N.I.M., Orban, D., Toint, P.L.: Cuter and sifdec: a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29(4), 373–394 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Gould, N.I.M., Sainvitu, C., Toint, P.L.: A filter-trust-region method for unconstrained optimization. SIAM J. Optim. 16(2), 341–357 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23(4), 707–716 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Grippo, L., Lampariello, F., Lucidi, S.: A truncated Newton method with nonmonotone line search for unconstrained optimization. J. Optim. Theory Appl. 60(3), 401–419 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn., Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)

    MATH  Google Scholar 

  16. 16.

    Nocedal, J., Yuan, Y.: Combining trust region and line search techniques. In: Advances in Nonlinear Programming, Beijing, 1996. Appl. Optim., vol. 14, pp. 153–175. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

  17. 17.

    Toint, P.L.: Non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints. Math. Program., Ser. A 77(1), 69–94 (1997)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Ulbrich, M., Ulbrich, S., Vicente, L.N.: A globally convergent primal-dual interior-point filter method for nonlinear programming. Math. Program., Ser. A 100(2), 379–410 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM J. Optim. 16(1), 1–31 (2005)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to N. Mahdavi-Amiri.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fatemi, M., Mahdavi-Amiri, N. A filter trust-region algorithm for unconstrained optimization with strong global convergence properties. Comput Optim Appl 52, 239–266 (2012). https://doi.org/10.1007/s10589-011-9411-5

Download citation

Keywords

  • Unconstrained optimization
  • Filter methods
  • Trust-region algorithms