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An Active Set-Type Newton Method for Constrained Nonlinear Systems

  • Christian Kanzow
Part of the Applied Optimization book series (APOP, volume 50)

Abstract

We consider the problem of finding a solution of a nonlinear system of equations subject to some box constraints. To this end, we introduce a new active set-type Newton method. This method is shown to be globally convergent in the sense that every accumulation point is a stationary point of a corresponding box constrained optimization problem. Moreover, the method is locally superlinearly or quadratically convergent under a suitable regularity condition. Furthermore the method generates feasible iterates and has to solve only one linear system of equations at each iteration. Due to our active set strategy, this linear system is of reduced dimension. Some preliminary numerical results are included.

Keywords

Nonlinear equations box constraints Newton’s method active set strategy projected gradient global convergence quadratic convergence 

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References

  1. [1]
    Bertsekas, D.P. (1982). Projected Newton methods for optimization problems with simple constraints. SIAM J. Control Optim., 20: 221–246.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Bertsekas, D.P. (1995). Nonlinear Programming. Athena Scientific, Belmont, MA.zbMATHGoogle Scholar
  3. [3]
    Calamai, P.H. and More, J.J. (1987). Projected gradient methods for linearly constrained problems. Math. Programming, 39: 93–116.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Dennis, Jr., J.E. and Schnabel, R.B. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ.zbMATHGoogle Scholar
  5. [5]
    Deuflhard, P. and Hohmann, A. (1995). Numerical Analysis. A First Course in Scientific Computation. Verlag de Gruyter, Berlin, New York.Google Scholar
  6. [6]
    Facchinei, F., Fischer, A. and Kanzow, C . (1997). A semismooth Newton method for variational inequalities: The case of box constraints. In Ferris, M.C. and Pang, J.-S., editors, Complementarity and Variational Problems: State of the Art, pages 76–90. SIAM, Philadelphia, PA.Google Scholar
  7. [7]
    Facchinei, F., Fischer, A. and Kanzow, C. (1999). On the accurate identification of active constraints. SIAM J. Optim., 9: 14–32.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Facchinei, F. and Soares, J. (1997). A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7: 225–247.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Ferris, M.C., Kanzow, C. and Munson, T.S. (1999). Feasible descent algorithms for mixed complementarity problems. Math. Programming, 86: 475–497.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Fletcher, R. (1987). Practical Methods of Optimization. John Wiley & Sons, New York, NY.zbMATHGoogle Scholar
  11. [11]
    Gafni, E.M. and Bertsekas, D.P. (1982). Convergence of a gradient projection method. Report LIDS-P-1201, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, MA.Google Scholar
  12. [12]
    Kanzow, C. (1999) Strictly feasible equation-based methods for mixed complementarity problems. Preprint 145, Institute of Applied Mathematics, University of Hamburg, Hamburg, Germany.Google Scholar
  13. [13]
    Kanzow, C. and Qi, H.-D. (1999). A QP-free constrained Newton-type method for variational inequality problems. Math. Programming, 85: 81–106.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Kelley, C.T. (1995). Iterative Methods for Linear and Nonlinear Equations., SIAM Philadelphia, PA.zbMATHCrossRefGoogle Scholar
  15. [15]
    Kelley, C.T. (1999): Iterative Methods for Optimization. SIAM, Philadelphia, PA.zbMATHCrossRefGoogle Scholar
  16. [16]
    Kozakevich, D.N., Martinez, J.M. and Santos, S.A. Solving nonlinear systems of equations with simple constraints. Computational and Applied Mathematics, to appear.Google Scholar
  17. [17]
    Lin, C-J. and More, J.J. (1999). Newton’s method for large bound-constrained optimization problems. SIAM J. Optim., 9: 1100–1127.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Liu, J. (1995). Strong stability in variational inequalities. SIAM J. Control Optim., 33: 725–749.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    More, J.J. and Sorensen, D.C (1983). Computing a trust region step. SIAM J. Sci. Stat. Comp., 4: 553–572.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Robinson, S.M. (1980). Strongly regular generalized equations. Math. Oper. Res., 5: 43–62.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Sun, D. (1998). Projected Newton-type methods for constrained nonsmooth equations. Talk presented at the “Workshop on Nons-mooth and Smoothing Methods” in Hong Kong, China.Google Scholar
  22. [22]
    Ulbrich, M. Non-monotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems. SIAM J. Optim., to appear.Google Scholar
  23. [23]
    Wang, T., Monteiro, R.D.C., and Pang, J.-S. (1996). An interior point potential reduction method for constrained equations. Math. Programming, 74: 159–195.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Christian Kanzow
    • 1
  1. 1.Institute of Applied MathematicsUniversity of HamburgHamburgGermany

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