A non-interior continuation algorithm for the P0 or P * LCP with strong global and local convergence properties

Applied Mathematics and Optimization

Volumn 52, Issue 2, 2005, Pages 237-262

  Huang, Zheng Hai ^a   Sun, Jie ^b  

^a TIANJIN UNIVERSITY   (China)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Global convergence; Global linear convergence; Linear complementarity problem; Local superlinear convergence; Non interior continuation algorithm

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; MATRIX ALGEBRA; PROBLEM SOLVING;

GLOBAL CONVERGENCE; GLOBAL LINEAR CONVERGENCE; LINEAR COMPLEMENTARITY PROBLEMS; LOCAL SUPERLINEAR CONVERGENCE; NON-INTERIOR CONTINUATION ALGORITHMS;

ALGORITHMS;

EID: 29544436758     PISSN: 00954616     EISSN: 14320606     Source Type: Journal    
DOI: 10.1007/s00245-005-0827-0     Document Type: Article

References (38)

1. Burke, J., and Xu, S., The global linear convergence of a non-interior path-following algorithm for linear complementarity problems, Math. Oper. Res., 23 (1998), 719-734.
2. Burke, J., and Xu, S., A non-interior predictor-corrector path-following algorithm for LCP, in Reformulation: Nonsmooth, Piecewise Smooth and Smoothing Methods, edited by Fukushima, M., and Qi, L., Kluwer, Boston, MA, pp. 45-64, 1999.
3. Burke, J., and Xu, S., A non-interior predictor-corrector path-following algorithm for the monotone linear complementarity problem, Math. Program., 87 (2000), 113-130.
4. 0 and monotone NCP, SIAM J. Optim., 9 (1999), 624-645.
5. Chen, B., and Chen, X., A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints, Comput. Optim. Appl., 13 (2000), 131-158.
6. Chen, B., and Harker, P.T., A non-interior-point continuation method for the linear complementarity problem, SIAM J. Matrix Anal. Appl., 14 (1993), 1168-1190.
7. Chen, B., and Xiu, N., A global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions, SIAM J. Optim., 9 (1999), 605-623.
8. Chen, B., and Xiu, N., Superlinear noninterior one-step continuation method for monotone LCP in the absence of strict complementarity, J. Optim. Theory Appl., 108 (2001), 317-332.
9. Chen, C., and Mangasarian, O.L., A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 5 (1996), 97-138.
10. Chen, X., and Tseng, P., Non-interior continuation methods for solving semidefinite complementarity problems, Math. Program., 95 (2003), 431-474.
11. 0 matrix linear complementarity problem, SIAM J. Optim., 11 (2000), 341-363.
12. Cottle, R.W., Pang, J.-S., and Stone, R.E., The Linear Complementarity Problem, Academic Press, Boston, MA, 1992.
13. Engelke, S., and Kanzow, C., Predictor-corrector smoothing methods for the solution of linear programming, Preprint, Department of Mathematics, University of Hamburg, March 2000.
14. Engelke, S., and Kanzow, C., Improved smoothing-type methods for the solution of linear programming, Numer. Math., 90 (2002), 487-507.
15. Fischer, A., A special Newton-type optimization method, Optim., 24 (1992), 269-284.
16. Gabriel, S.A., and More, J.J., Smoothing of mixed complementarity problems, in Complementarity and Variational Problems: State of the Art, edited by Ferris, M.C., and Pang, J.-S., SIAM, Philadelphia, PA, pp. 105-116, 1997.
17. 0-equations, Comput. Optim. Appl., 12 (1999), 229-251.
18. Hotta, K., and Yoshise, A., Global convergence of non-interior-point algorithms using Chen-HarkerKanzow functions for nonlinear complementarity problems, Math. Program., 86 (1999), 105-133.
19. 0 function nonlinear complementarity problem, Oper. Res. Lett., 30 (2002), 202-210.
20. Huang, Z.H., The global linear and local quadratic convergence of a non-interior continuation algorithm for the LCP. IMA J. Numer. Anal., to appear.
21. Huang, Z.H., Locating a maximally complementary solution of the monotone NCP by using non-interior-point smoothing algorithms, Math. Methods Oper. Res., 61 (2005), 41-55.
22. Huang, Z.H., and Han, J., Non-interior continuation method for solving the monotone semidefinite complementarity problem, Appl. Math. Optim., 47 (2003), 195-211.
23. 0 function, J. Optim. Theory Appl., 117 (2003), 39-68.
24. 0- and monotone LCP, Math. Program., 99 (2004), 423-441.
25. Kanzow, C., Some noninterior continuation methods for linear complementarity problems, SIAM J. Matrix Anal. Appl., 17 (1996), 851-868.
26. Kanzow, C., A new approach to continuation methods for complementarity problems with P-functions, Oper. Res. Lett., 20 (1997), 851-868.
27. Kojima, M., Megiddo, N., Noma, T., and Yoshise, A., A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Sciences, Vol. 538, Springer-Verlag, Berlin, 1991.
28. Mifflin, R., Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15(1977), 957-972.
29. Qi, L., Convergence analysis of some algorithm for solving nonsmooth equations, Math. Oper. Res., 18 (1993), 227-244.
30. Qi, L., and Sun, J., A nonsmooth version of Newton's method, Math. Program., 58 (1993), 353-367.
31. Qi, L., and Sun, D., Improving the convergence of non-interior point algorithm for nonlinear complementarity problems, Math. Comput., 69 (2000), 283-304.
32. Smale, S., Algorithms for solving equations, in Proceeding of the International Congress of Mathematicians, edited by Gleason, A.M., American Mathematical Society, Providence, Rl, pp. 172-195, 1987.
33. Sun, J., and Huang, Z.H., Finite termination of a smoothing Newton algorithm for solving the LCP with a sufficient matrix, Technical Report, Dept. of Decision Sciences, National University of Singapore, 2003.
34. Tseng, P., Analysis of a non-interior continuation method based on Chen-Mangasarian smoothing functions for complementarity problems, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, edited by Fukushima, M., and Qi, L., Kluwer, Boston, MA, pp. 381-404, 1999.
35. Tseng, P., Error bounds and superlinear convergence analysis of some Newton-type methods in optimization, in Nonlinear Optimization and Related Topics, edited by Di Pillo, G., and Giannessi, F., Kluwer, Boston, MA, pp. 445-462, 2000.
36. Xu, S., The global linear convergence of an infeasible non-interior path-following algorithm for complementarity problems with uniform P-functions, Math. Program., 87 (2000), 501-517.
37. Zhao, Y., and Li, D., Existence and limiting behavior of a non-interior-point trajectory for nonlinear complementarity problems without strict feasibility conditions, SIAM J. Control Optim., 40 (2001), 898-924.
38. 0 LCPs, SIAMJ. Optim., 13 (2003), 1196-1221.