Skip to main content
SpringerLink
Log in

An interior-point affine-scaling trust-region method for semismooth equations with box constraints

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

Abstract

An algorithm for the solution of a semismooth system of equations with box constraints is described. The method is an affine-scaling trust-region method. All iterates generated by this method are strictly feasible. In this way, possible domain violations outside or on the boundary of the box are avoided. The method is shown to have strong global and local convergence properties under suitable assumptions, in particular, when the method is used with a special scaling matrix. Numerical results are presented for a number of problems arising from different areas.

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. Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. Wiley, New York (1979)

    Google Scholar 

  2. Bellavia, S., Macconi, M., Morini, B.: An affine scaling trust-region approach to bound-constrained nonlinear systems. Appl. Numer. Math. 44, 257–280 (2003)

    Article  MathSciNet  Google Scholar 

  3. Bellavia, S., Macconi, M., Morini, B.: STRSCNE: a scaled trust-region solver for constrained nonlinear systems. Comput. Optim. Appl. 28, 31–50 (2004)

    Article  MathSciNet  Google Scholar 

  4. Bellavia, S., Macconi, M., Morini, B.: A two-dimensional trust-region method for large scale bound-constrained nonlinear systems. Technical report, submitted for publication

  5. Bellavia, S., Morini, B.: An interior global method for nonlinear systems with simple bounds. Optim. Methods Softw. 20, 453–474 (2005)

    Article  MathSciNet  Google Scholar 

  6. Bertsekas, D.P.: Projected Newton methods for optimization problems with simple constraints. SIAM J. Control Optim. 20, 221–246 (1982)

    Article  MathSciNet  Google Scholar 

  7. Byrd, R.H., Lu, P., Nocedal, J.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Stat. Comput. 16, 1190–1208 (1995)

    Article  MathSciNet  Google Scholar 

  8. Chen, B., Chen, X., Kanzow, C.: A penalized Fischer–Burmeister NCP-function. Math. Program. 88, 211–216 (2000)

    Article  MathSciNet  Google Scholar 

  9. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  Google Scholar 

  11. Coleman, T.F., Li, Y.: An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6, 418–445 (1996)

    Google Scholar 

  12. Conn, A.R., Gould, N.I.M., Toint, P.L.: Global convergence of a class of trust region algorithms for optimization with simple bounds. SIAM J. Numer. Anal. 25, 433–460 (1998) (Correction in SIAM J. Numer. Anal. 26, 764–767 (1989))

    MathSciNet  Google Scholar 

  13. Conn, A.R., Gould, N.I.M., Toint, Ph.L.: Testing a class of methods for solving minimization problems with simple bounds on the variables. Math. Comput. 50, 399–430 (1988)

    MathSciNet  Google Scholar 

  14. Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. MPS/SIAM Series on Optimization. SIAM, Philadelphia (2000)

    MATH  Google Scholar 

  15. De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75, 407–439 (1996)

    Article  Google Scholar 

  16. Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  18. Dennis, J.E., Vicente, L.N.: Trust-region interior-point algorithms for minimization problems with simple bounds. In: Fischer, H., Riedmüller, B., Schäffler, S. (eds.) Applied Mathematics and Parallel Computing. Festschrift for Klaus Ritter, pp. 97–107. Physica, Heidelberg (1996)

    Google Scholar 

  19. Deuflhard, P.: Newton Methods for Nonlinear Problems. Springer, Berlin/Heidelberg (2004)

    MATH  Google Scholar 

  20. Dirkse, S.P., Ferris, M.C.: MCPLIB: A collection of nonlinear mixed complementarity problems. Optim. Methods Softw. 5, 319–345 (1995)

    Google Scholar 

  21. Facchinei, F., Fischer, A., Kanzow, C.: Inexact Newton methods for semismooth equations with applications to variational inequality problems. In: Di Pillo, G., Giannessi, F. (eds.) Nonlinear Optimization and Applications, pp. 125–139. Plenum, New York (1996)

    Google Scholar 

  22. Facchinei, F., Júdice, J., Soares, J.: An active set Newton algorithm for large-scale nonlinear programs with box constraints. SIAM J. Optim. 8, 158–186 (1998)

    Google Scholar 

  23. Facchinei, F., Lucidi, S., Palagi, L.: A truncated Newton algorithm for large scale box constrained optimization. SIAM J. Optim. 12, 1100–1125 (2002)

    Article  MathSciNet  Google Scholar 

  24. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. II. Springer, New York (2003)

    MATH  Google Scholar 

  25. Facchinei, F., Soares, J.: A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7, 225–247 (1997)

    Article  MathSciNet  Google Scholar 

  26. Ferris, M.C., Pang, J.-S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)

    Article  MathSciNet  Google Scholar 

  27. Floudas, C.A., et al.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic, Dordrecht (1999)

    MATH  Google Scholar 

  28. Friedlander, A., Martínez, J.M., Santos, S.A.: A new trust region algorithm for bound constrained minimization. Appl. Math. Optim. 30, 235–266 (1994)

    Article  Google Scholar 

  29. 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)

    Article  MathSciNet  Google Scholar 

  30. Kanzow, C.: Strictly feasible equation-based methods for mixed complementarity problems. Numer. Math. 89, 135–160 (2001)

    Article  MathSciNet  Google Scholar 

  31. Kanzow, C.: An active set-type Newton method for constrained nonlinear systems. In: Ferris, M.C., Mangasarian, O.L., Pang, J.-S. (eds.) Complementarity: Applications, Algorithms and Extensions, pp. 179–200. Kluwer Academic, Dordrecht (2001)

    Google Scholar 

  32. Kanzow, C., Klug, A.: On affine-scaling interior-point Newton methods for nonlinear minimization with bound constraints. Comput. Optim. Appl. 35, 177–197 (2006)

    Article  MathSciNet  Google Scholar 

  33. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)

    MATH  Google Scholar 

  34. Kelley, C.T.: Solving Nonlinear Equations with Newton’s Method. SIAM, Philadelphia (2003)

    MATH  Google Scholar 

  35. Kozakevich, D.N., Martínez, J.M., Santos, S.A.: Solving nonlinear systems of equations with simple constraints. Comput. Appl. Math. 16, 215–235 (1997)

    Google Scholar 

  36. Lescrenier, M.: Convergence of trust region algorithms for optimization with bounds when strict complementarity does not hold. SIAM J. Numer. Anal. 28, 476–495 (1991)

    Article  MathSciNet  Google Scholar 

  37. Lewis, R.M., Torczon, V.: Pattern search algorithms for bound constrained minimization. SIAM J. Optim. 9, 1082–1099 (1999)

    Article  MathSciNet  Google Scholar 

  38. Lin, C.-J., Moré, J.J.: Newton’s method for large bound-constrained optimization problems. SIAM J. Optim. 9, 1100–1127 (1999)

    Article  Google Scholar 

  39. Lukšan, L.: Inexact trust region method for large sparse systems of nonlinear equations. J. Optim. Theory Appl. 81, 569–590 (1994)

    Google Scholar 

  40. Martínez, J.M., Qi, L.: Inexact Newton methods for solving nonsmooth equations. J. Comput. Appl. Math. 60, 127–145 (1995)

    Google Scholar 

  41. Meintjes, K., Morgan, A.P.: Chemical equilibrium systems as numerical test problems. ACM Trans. Math. Softw. 16, 143–151 (1990)

    Article  Google Scholar 

  42. Moré, J.J., Cosnard, M.Y.: Numerical solution of nonlinear equations. ACM Trans. Math. Softw. 5, 64–85 (1979)

    Article  Google Scholar 

  43. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic, New York/London (1970)

    MATH  Google Scholar 

  44. Pang, J.-S., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993)

    Article  MathSciNet  Google Scholar 

  45. Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    Article  Google Scholar 

  46. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)

    Article  MathSciNet  Google Scholar 

  47. Qi, L., Tong, X., Li, D.: Active-set projected trust region algorithm for box constrained nonsmooth equations. J. Optim. Theory Appl. 120, 601–625 (2004)

    Article  MathSciNet  Google Scholar 

  48. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    MATH  Google Scholar 

  49. Schwartz, A., Polak, E.: Family of projected descent methods for optimization problems with simple bounds. J. Optim. Theory Appl. 92, 1–31 (1997)

    Article  MathSciNet  Google Scholar 

  50. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 2nd edn. Springer, New York (1993)

    MATH  Google Scholar 

  51. Tong, X.J., Qi, L.: On the convergence of a trust-region method for solving constrained nonlinear equations with degenerate solutions. J. Optim. Theory Appl. 123, 187–211 (2004)

    Article  MathSciNet  Google Scholar 

  52. Ulbrich, M.: Non-monotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems. SIAM J. Optim. 11, 889–917 (2001)

    Article  MathSciNet  Google Scholar 

  53. Ulbrich, M., Ulbrich, S.: Superlinear convergence of affine-scaling interior-point Newton methods for infinite-dimensional nonlinear problems with pointwise bounds. SIAM J. Control Optim. 38, 1938–1984 (2000)

    Article  MathSciNet  Google Scholar 

  54. Ulbrich, M., Ulbrich, S., Heinkenschloss, M.: Global convergence of trust-region interior-point algorithms for infinite-dimensional nonconvex minimization subject to pointwise bounds. SIAM J. Control Optim. 37, 731–764 (1999)

    Article  MathSciNet  Google Scholar 

  55. Zhu, C., Byrd, R.H., Nocedal, J.: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization. ACM Trans. Math. Softw. 23, 550–560 (1997)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Würzburg, Am Hubland, 97074, Würzburg, Germany

    Christian Kanzow & Andreas Klug

Authors
  1. Christian Kanzow
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Klug
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Kanzow.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kanzow, C., Klug, A. An interior-point affine-scaling trust-region method for semismooth equations with box constraints. Comput Optim Appl 37, 329–353 (2007). https://doi.org/10.1007/s10589-007-9029-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-007-9029-9

Keywords

Search

Navigation