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

Computational Optimization and Applications

Volumn 37, Issue 3, 2007, Pages 329-353

  Kanzow, Christian ^a   Klug, Andreas ^a  

^a UNIVERSITY OF WÜRZBURG   (Germany)

Author keywords

Affine scaling; Box constraints; Newton's method; Nonlinear equations; Semismooth functions; Trust region method

Indexed keywords

BOUNDARY CONDITIONS; CONSTRAINT THEORY; CONVERGENCE OF NUMERICAL METHODS; GLOBAL OPTIMIZATION; NEWTON-RAPHSON METHOD;

AFFINE SCALING; BOX CONSTRAINTS; SEMISMOOTH FUNCTIONS; TRUST-REGION METHODS;

NONLINEAR EQUATIONS;

EID: 34547367209     PISSN: 09266003     EISSN: 15732894     Source Type: Journal    
DOI: 10.1007/s10589-007-9029-9     Document Type: Article

References (55)

1. Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. Wiley, New York (1979)
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)
3. Bellavia, S., Macconi, M., Morini, B.: STRSCNE: a scaled trust-region solver for constrained nonlinear systems. Comput. Optim. Appl. 28, 31-50 (2004)
4. Bellavia, S., Macconi, M., Morini, B.: A two-dimensional trust-region method for large scale boundconstrained 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, 453-414 (2005)
6. Bertsekas, D.P.: Projected Newton methods for optimization problems with simple constraints. SIAM J. Control Optim. 20, 221-246 (1982)
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)
8. Chen, B., Chen, X., Kanzow, C.: A penalized Fischer-Burmeister NCP-function. Math. Program. 88, 211-216 (2000)
9. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
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)
11. Coleman, T.F., Li, Y.: An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6, 418-445 (1996)
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))
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)
14. Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. MPS/SIAM Series on Optimization. SIAM, Philadelphia (2000)
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)
16. Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400-408 (1982)
17. Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs (1983)
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)
19. Deuflhard, P.: Newton Methods for Nonlinear Problems. Springer, Berlin/Heidelberg (2004)
20. Dirkse, S.P., Ferris, M.C.: MCPLIB: A collection of nonlinear mixed complementarity problems. Optim. Methods Softw. 5, 319-345 (1995)
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)
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)
23. Facchinei, F., Lucidi, S., Palagi, L.: A truncated Newton algorithm for large scale box constrained optimization. SIAM J. Optim. 12, 1100-1125 (2002)
24. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. II. Springer, New York (2003)
25. Facchinei, F., Soares, J.: A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7, 225-247 (1997)
26. Ferris, M.C., Pang, J.-S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669-713 (1997)
27. Floudas, C.A., et al.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic, Dordrecht (1999)
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)
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)
30. Kanzow, C.: Strictly feasible equation-based methods for mixed complementarity problems. Numer. Math. 89, 135-160 (2001)
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)
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)
33. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)
34. Kelley, C.T.: Solving Nonlinear Equations with Newton's Method. SIAM, Philadelphia (2003)
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)
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)
37. Lewis, R.M., Torczon, V.: Pattern search algorithms for bound constrained minimization. SIAM J. Optim. 9, 1082-1099 (1999)
38. Lin, C.-J., Moré, J.J.: Newton's method for large bound-constrained optimization problems. SIAM J. Optim. 9, 1100-1127 (1999)
39. Lukšan, L.: Inexact trust region method for large sparse systems of nonlinear equations. J. Optim. Theory Appl. 81, 569-590 (1994)
40. Martínez, J.M., Qi, L.: Inexact Newton methods for solving nonsmooth equations. J. Comput. Appl. Math. 60, 127-145 (1995)
41. Meintjes, K., Morgan, A.P.: Chemical equilibrium systems as numerical test problems. ACM Trans. Math. Softw. 16, 143-151 (1990)
42. More, J.J., Cosnard, M.Y.: Numerical solution of nonlinear equations. ACM Trans. Math. Softw. 5, 64-85 (1979)
43. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic, New York/London (1970)
44. Pang, J.-S., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443-465 (1993)
45. Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227-244 (1993)
46. Qi, L., Sun, J.: A nonsmooth version of Newton's method. Math. Program. 58, 353-367 (1993)
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)
48. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)
49. Schwartz, A., Polak, E.: Family of projected descent methods for optimization problems with simple bounds. J. Optim. Theory Appl. 92, 1-31 (1997)
50. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 2nd edn. Springer, New York (1993)
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)
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)
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)
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)
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)