Global linear and local quadratic continuation smoothing method for variational inequalities with box constraints

Computational Optimization and Applications

Volumn 17, Issue 2-3, 2000, Pages 131-158

  Chen, Bintong ^a   Chen, Xiaojun ^b  

^a WASHINGTON STATE UNIVERSITY   (United States)

^b SHIMANE UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; CONSTRAINT THEORY; CONVERGENCE OF NUMERICAL METHODS; ITERATIVE METHODS; LINEAR EQUATIONS; PROBLEM SOLVING;

BOX CONSTRAINTS; CONTINUATION SMOOTHING METHOD; LINEAR CONVERGENCE; QUADRATIC CONVERGENCE; VARIATIONAL INEQUALITIES;

VARIATIONAL TECHNIQUES;

EID: 0034476547     PISSN: 09266003     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1026546230851     Document Type: Article

References (44)

1. S.C. Billups, S.P. Dirkse, and M.C. Ferris, "A comparison of large-scale mixed complementarity problems solvers," Comp. Optim. Appl., vol. 7, pp. 3-25, 1997.
2. J. Burke and S. Xu, "The global linear convergence of a non-interior path-following algorithm for linear complementarity problem," Math. of Oper. Res., vol. 23, pp. 719-734, 1998.
3. B. Chen, "Finite convergence of nonsmooth equation based methods for affine variational inequalities," Appl. Math. Lett., vol. 5, pp. 19-24, 1992.
4. B. Chen and P.T. Harker, "A non-interior-point continuation method for linear complementarity problems," SIAM J. Matrix Anal. Appl., vol. 14, pp. 1168-1190, 1993.
5. B. Chen and P.T. Harker, "A non-interior-point continuation method for quadratic and linear programs," SIAM J. Optim., vol. 3, pp. 503-515, 1993.
6. B. Chen and P.T. Harker, "A continuation method for monotone variational inequalities," Math. Programming, vol. 69, pp. 237-253, 1995.
7. B. Chen and P.T. Harker, "Smooth approximations to nonlinear complementarity problems," SIAM J. Optim., vol. 7, pp. 403-420, 1997.
8. B. Chen and N. Xiu, "A global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing function," SIAM J. Optim., vol. 9, pp. 605-623, 1999.
9. C. Chen and O.L. Mangasarian, "Smoothing methods for convex inequalities and linear complementarity problems," Math. Programming, vol. 71, pp. 51-69, 1995.
10. C. Chen and O.L. Mangasarian, "A class of smoothing functions for nonlinear and mixed complementarity problems," Comp. Optim. and Appl., vol. 5, pp. 97-138, 1996.
11. X. Chen and L. Qi, "A parameterized Newton method and a Broyden-like method for solving nonsmooth equations," Comp. Optim. Appl., vol. 3, pp. 157-179, 1994.
12. X. Chen, L. Qi, and D. Sun, "Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities," Math. of Comp., vol. 67, pp. 519-540, 1998.
13. X. Chen and Y. Ye, "On homotopy-smoothing methods for variational inequalities," SIAM J. Con. and Optim., vol. 37, pp. 589-616, 1999.
14. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley: New York, 1983.
15. B.C. Eaves, "On the basic theorem of complementarity," Math. Programming, vol. 1, pp. 68-75, 1971.
16. M.C. Ferris and J.S. Pang, "Engineering and economic applications of complementarity problems," SIAM Rev., vol. 39, pp. 669-713, 1997.
17. F. Facchinei, A. Fischer, and C. Kanzow, "A semismooth Newton method for variational inequalities: The case of box constraints," in Complementarity and Variational Problems: State of the Art, M.C. Ferris and J.S. Pang (Eds.), SIAM: Philadelphia, Pennsylvania, 1996.
18. H. Jiang, "Global convergence analysis of the generalized newton and gauss-newton methods of the Fischer-Burmeister equation for the complementarity problem," Math. of Oper. Res., vol. 24, pp. 529-543, 1999.
19. A. Fischer and C. Kanzow, "On the finite termination of an iterative method for linear complementarity problems," Math. Programming, vol. 74, pp. 279-292, 1996.
20. R.M. Freund and S. Mizuno, "Interior point methods: current status and future directions," OPTIMA Newsletter, vol. 51, pp. 1-9, 1996.
21. S.A. Gabriel, "A hybrid smoothing method for mixed nonlinear complementarity problems," Comp. Optim. and Appl., vol. 9, pp. 153-173, 1998.
22. S.A. Gabriel and J.J. More, "Smoothing of mixed complementarity problems," in Complementarity and Variational Problems: State of the Art, M.C. Ferris and J.S. Pang, (Eds.), SIAM: Philadelphia, Pennsylvania, 1997, pp. 105-116.
23. P.T. Harker and J.S. Pang, "Finite-dimensional variational inequality and nonlinear complementarity problem: A survey of theory, algorithms and applications," Math. Programming, vol. 48, pp. 339-357, 1990.
24. K. Hotta and A. Yoshise, "Global convergence of a class of non-interior-point algorithms using Chen-Harker-Kanzow functions for nonlinear complementarity problems," Math. Programming, vol. 86, pp. 105-133, 1999.
25. C. Kanzow, "Some noninterior continuation methods for linear complementarity problems," SIAM J. Matrix Anal. Appl., vol. 17, pp. 851-868, 1996.
26. C. Kanzow and M. Fukushima, "Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities," Math. Programming, vol. 83, pp. 55-87, 1998.
27. C. Kanzow and H. Jiang, "A continuation method for (strongly) monotone variational inequalities," Math. Programming, vol. 81, pp. 140-157, 1998.
28. M. Kojima, N. Megiddo, and S. Mizuno, "A general framework of continuation methods for complementarity problems," Math. of Oper. Res., vol. 18, pp. 945-963, 1993.
29. M. Kojima, N. Megiddo, and T. Noma, "Homotopy continuation methods for nonlinear complementarity problems," Math. of Oper. Res., vol. 16, pp. 754-774, 1991.
30. J.S. Pang, "Newton's method for B-differentiable equations," Math. Oper. Res., vol.15, pp. 311-341, 1990.
31. J.S. Pang, "Complementarity problems," in Handbook of Global Optimization, R. Horst and P. Pardalos (Eds.), Kluwer Academic Publishers: Boston, 1995, pp. 271-338.
32. L. Qi, "Convergence analysis of some algorithms for solving nonsmooth equations," Math. Oper. Res., vol. 18, pp. 227-244, 1993.
33. L. Qi and X. Chen, "A globally convergent successive approximation method for severely nonsmooth equations," SIAM J. Control Optim., vol. 33, pp. 402-418, 1995.
34. L. Qi and J. Sun, "A nonsmooth version of Newton's method," Math. Programming, vol. 58, pp. 353-367, 1993.
35. L. Qi, D. Sun, and G. Zhou, "A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities," Math. Programming, vol. 87, pp. 1-35, 2000.
36. H. Sellami and S.M. Robinson, "Implementation of a continuation method for normal maps," Math. Programming, vol. 76, pp. 563-578, 1997.
37. S. Smale, "Algorithms for solving equations," in Proceedings of the International Congress of Mathematicians, Berkeley, California, 1986, pp. 172-195.
38. D. Sun and R.S. Womersley, "A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Guass-Newton method," SIAM J. Optim., vol. 9, pp. 388-413, 1999.
39. P. Tseng, "An infeasible path-following method for monotone complementarity problem," SIAM J. Optim., vol. 7, pp. 386-402, 1997.
40. S. Wright and D. Ralph, "A superlinear infeasible-interior-point algorithm for monotone complementarity problems," Math. Oper. Res., vol. 21, pp. 815-838, 1996.
41. S. Xu, "The global linear convergence of an infeasible non-interior path-following algorithm for complementarity problems with uniform P-functions," Math. Programming, vol. 87, pp. 501-517, 2000.
42. S. Xu, "The global linear convergence and complexity of a non-interior path-following algorithm for monotone LCP based on Chen-Harker-Kanzow-Smale smooth functions," Department of Mathematics, University of Washington, Seattle, WA 98195, Preprint, 1997.
43. Y. Ye and K. Anstreicher, "On quadratic and O(√N L) convergence of a predictor-corrector algorithm for LCP," Math. Programming, vol. 62, pp. 537-552, 1993.
44. I. Zang, "A smoothing-out technique for min-max optimization," Math. Programming, vol. 19, pp. 61-71, 1980.