# An Active Set-Type Newton Method for Constrained Nonlinear Systems

• Christian Kanzow
Chapter
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

## References

1. [1]
Bertsekas, D.P. (1982). Projected Newton methods for optimization problems with simple constraints. SIAM J. Control Optim., 20: 221–246.
2. [2]
Bertsekas, D.P. (1995). Nonlinear Programming. Athena Scientific, Belmont, MA.
3. [3]
Calamai, P.H. and More, J.J. (1987). Projected gradient methods for linearly constrained problems. Math. Programming, 39: 93–116.
4. [4]
Dennis, Jr., J.E. and Schnabel, R.B. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ.
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.
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.
9. [9]
Ferris, M.C., Kanzow, C. and Munson, T.S. (1999). Feasible descent algorithms for mixed complementarity problems. Math. Programming, 86: 475–497.
10. [10]
Fletcher, R. (1987). Practical Methods of Optimization. John Wiley & Sons, New York, NY.
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.
14. [14]
Kelley, C.T. (1995). Iterative Methods for Linear and Nonlinear Equations., SIAM Philadelphia, PA.
15. [15]
Kelley, C.T. (1999): Iterative Methods for Optimization. SIAM, Philadelphia, PA.
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.
18. [18]
Liu, J. (1995). Strong stability in variational inequalities. SIAM J. Control Optim., 33: 725–749.
19. [19]
More, J.J. and Sorensen, D.C (1983). Computing a trust region step. SIAM J. Sci. Stat. Comp., 4: 553–572.
20. [20]
Robinson, S.M. (1980). Strongly regular generalized equations. Math. Oper. Res., 5: 43–62.
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.