A regularization Newton method for solving nonlinear complementarity problems

Applied Mathematics and Optimization

Volumn 40, Issue 3, 1999, Pages 315-339

  Sun, D ^a  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

Convergence; Generalized Newton method; Nonlinear complementarity problem; Nonsmooth equations; Regularization

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; PROBLEM SOLVING; THEOREM PROVING;

NONLINEAR COMPLEMENTARITY PROBLEMS; NONSMOOTH EQUATIONS; REGULARIZATION NEWTON METHOD;

ITERATIVE METHODS;

EID: 0032682238     PISSN: 00954616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002459900128     Document Type: Article

References (47)

1. 0 and monotone NCP, Preprint, Department of Management and Systems, Washington State University, Pullman, WA, May 1997. Revised November 1997. SIAM Journal on Optmization, to appear.
2. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
3. R.W. Cottle, J.-S. Pang, and R.E. Stone, The Linear Complementarity Problem, Academic Press, Boston, 1992.
4. T. De Luca, F. Facchinei, and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Mathematical Programming, 75 (1996), 407-439.
5. A.L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Lecture Notes in Mathematics 1543, Springer-Verlag, Berlin, 1993.
6. J. Eckstein and M.C. Ferris, Smooth methods of multipliers for complementarity problems, RUCTOR Research Report 27-96, Faculty of Management and RUTCOR, Rutgers University, New Brunswick, NJ 08903, February 1997.
7. 0 nonlinear complementarity problems, Mathematics of Operations Research, 23 (1998), 735-745.
8. F. Facchinei, A. Fischer, and C. Kanzow, Regularity properties of a semismooth equation reformulation of variational inequalities, SIAM Journal on Optimization, 8 (1998), 850-869.
9. F. Facchinei and C. Kanzow, Beyond monotonicity in regularization methods for nonlinear complementarity problems, DIS Working Paper 11-97, Università di Roman "La Sapienza," Rome, May 1997. SIAM Journal on Control and Optimization, to appear.
10. F. Facchinei and J. Soares, A new merit function for nonlinear complementarity problems and a related algorithm, SIAM Journal on Optimization, 7 (1997), 225-247.
11. M.C. Ferris and J.-S. Pang, Engineering and economic applications of complementarity problems, SIAM Review, 39 (1997), 669-713.
12. A. Fischer, A special Newton-type optimization method, Optimization, 24 (1992), 269-284.
13. A. Fischer, An NCP-function and its use for the solution of complementarity problems, in Recent Advances in Nonsmooth Optimization, D. Du, L. Qi, and R. Womersley, eds., World Scientific, NJ, pp. 88-105, 1995.
14. A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions, Mathematical Programming, 76 (1997), 513-532.
15. M. Fukushima, Merit functions for variational inequality and complementarity problems, in Nonlinear Optimization and Applications, G. Di Pillo and F. Giannessi, eds., Plenum, New York, pp. 155-170, 1996.
16. M.S. Gowda and R. Sznajder, "Weak univalence and connectedness of inverse images of continuous functions", Preprint, Department of Mathematics & Statistics, University of Maryland Baltimore County, Baltimore, MD 21250, April 1997.
17. P.T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications, Mathematical Programming, 48 (1990), 161-220.
18. K. Hotta and A. Yoshise, Global convergence of a class of non-interior-point algorithms using Chen-Harker-Kanzow functions for nonlinear complementarity problems, Discussion Paper Series No. 708, Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Ibaraki 305, December 1996.
19. H. Jiang, Smoothed Fischer-Burmeister equation methods for the nonlinear complementarity problem, Preprint, Department of Mathematics, University of Melbourne, Victoria 3052, June 1997.
20. H. Jiang and L. Qi, A new nonsmooth equation approach to nonlinear complementarity problems, SIAM Journal on Control and Optimization, 35 (1997), 178-193.
21. C. Kanzow, An unconstrained optimization technique for large-scale linearly constrained convex minimization problems, Computing, 53 (1994), 101-117.
22. C. Kanzow, Global convergence properties of some iteration methods for linear complementarity problems, SIAM Journal on Optimization, 6 (1996), 326-341.
23. L. Mathiesen, An algorithm based on a sequence of linear complementarity problems applied to a Walrasian equilibrium model: an example, Mathematical Programming, 37 (1987), 1-18.
24. R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM Journal on Control and Optimization, 15 (1977), 957-972.
25. J.-S. Pang, Complementarity problems, in Handbook of Global Optimization, R. Horst and P. Pardalos, eds., Kluwer, Boston, pp. 271-338, 1995.
26. J.-S. Pang and S.A. Gabriel, NE/SQP: a robust algorithm for the nonlinear complementarity problem, Mathematical Programming, 60 (1993), 295-337.
27. B.T. Polyak, Introduction to Optimization, Optimization Software Inc., New York, 1987.
28. L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, 18 (1993), 227-244.
29. L. Qi, Regular pseudo-smooth NCP and BVIP functions and globally and quadratically convergent generalized Newton methods for complementarity and variational inequality problems, Applied Mathematics Report AMR 97/14, School of Mathematics, University of New South Wales, Sydney 2052, NSW, June 1997. To appear in Mathematics of Operations Research.
30. L. Qi and H. Jiang, Semismooth Karush-Kuhn-Tucker equations and convergence analysis of Newton and quasi-Newton methods for solving these equations, Mathematics of Operations Research, 22 (1997), 301-325.
31. L. Qi and D. Sun, Improving the convergence of non-interior point algorithms for nonlinear complementarity problems, Applied Mathematics Report AMR 97/11, School of Mathematics, University of New South Wales, Sydney 2052, NSW, May 1997. Revised February 1998. Mathematics of Computation, to appear.
32. L. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Applied Mathematics Report AMR 97/13, School of Mathematics, University of New South Wales, Sydney 2052, NSW, June 1997.
33. L. Qi and J. Sun, A nonsmooth version of Newton's method, Mathematical Programming, 58 (1993), 353-367.
34. S.M. Robinson, Generalized equations, in Mathematical Programming: The State of the Art, A. Bachem, M. Grötschel and B. Korte, eds., Springer-Verlag, Berlin, pp. 346-347, 1983.
35. R.T. Rockafellar, Monotone operators and the proximal point algorithms, SIAM Journal on Control and Optimization, 14 (1976), 877-898.
36. R.T. Rockafellar, Monotone operators and augmented Lagrangian methods in nonlinear programming, in Nonlinear Programming 3, O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, eds., Academic Press, New York, pp. 1-25, 1978.
37. P.K. Subramanian, A note on least two norm solutions of monotone complementarity problems, Technical Report 2844, Mathematical Research Center, University of Wisconsin, Madison, WI, July 1985.
38. D. Sun and R.S. Womersley, A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Gauss-Newton method, Applied Mathematics Report AMR 96/37, School of Mathematics, University of New South Wales, Sydney 2052, NSW, December 1996. Revised September 1997. SIAM Journal on Optimization, to appear.
39. P. Tseng, Global behaviour of a class of merit functions for the nonlinear complementarity problem, Journal of Optimization Theory and Applications, 89 (1996), 17-37.
40. P. Tseng, An infeasible path-following method for the monotone complementarity problem", SIAM Journal on Optimization, 7 (1997), 386-402.
41. 0-matrix, SIAM Journal on Matrix Analysis and Applications, 14 (1993), 967-977.
42. L.T. Watson, Solving the nonlinear complementarity problem by a homotopy method, SIAM Journal on Control and Optimization, 17 (1979), 36-46.
43. S. Wright, Primal-Dual Interior Point Methods, SIAM, Philadelphia, PA, 1997.
44. S. Wright and D. Ralph, A superlinear infeasible-interior-point algorithm for monotone complementarity problems, Mathematics of Operations Research, 21 (1996), 815-838.
45. B. Xiao and PT. Harker, A nonsmooth Newton method for variational inequalities, II: Numerical results, Mathematical Programming, 65 (1994), 195-216.
46. N. Yamashita and M. Fukushima, Modified Newton methods for solving semismooth reformulations of monotone complementarity problems, Mathematical Programming, 76 (1997), 469-491.
47. E.H. Zarantonello, Projections on convex sets in Hubert space and spectral theory, in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, ed., Academic Press, New York, pp. 237-424, 1971.