Existence and Limiting Behavior of Trajectories Associated with P0-equations

Computational Optimization and Applications

Volumn 12, Issue 1-3, 1999, Pages 229-251

  Gowda, M Seetharama ^a   Tawhid, M A ^a  

^a UNIVERSITY OF MARYLAND BALTIMORE COUNTY   (United States)

Author keywords

Box variational inequality problem; Coercive; Fixed point map; Interior point trajectory; Min function; Nonlinear complementarity problem; Normal map; P0 function; Trajectory

Indexed keywords

EID: 0000742219     PISSN: 09266003     EISSN: None     Source Type: Journal    
DOI: 10.1007/978-1-4615-5197-3_12     Document Type: Article

References (57)

1. S. Banach and S. Mazur, "Uber Mehrdeutige Stetige Abildungen," Studia Math, vol. 5, pp. 174-178, 1934.
2. M.S. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977.
3. R. Benedetti and J.J. Risler, Real Algebraic and Semi-Algebraic Sets, Hermann, Éditeurs des Sciences et des Arts, Paris, 1990.
4. S.C. Billups and M.C. Ferris, "QPCOMP: A Quadratic Programming Based Solver for Mixed Complementarity Problems," Mathematical Programming, vol. 76, pp. 533-562, 1997.
5. J. Burke and S. Xu, "The Global Linear Convergence of a Non-Interior Path-Following Algorithm for Linear Complementarity Problems," Technical Report, Department of Mathematics, University of Washington, Seattle, WA, 1996 (revised July 1997).
6. 0 and Monotone NCP," SIAM J. Optimization, to appear.
7. B. Chen and X. Chen, "A Global Linear and Local Quadratic Continuation Smoothing Method for Variational Inequalities with Box Constraints," Technical Report, Department of Management and Systems, Washington State University, Pullman, WA, 1997.
8. B. Chen, X. Chen and C. Kanzow, "A Penalized Fischer-Burmeister NCP-Function: Theoretical Investigation and Numerical Results," Preprint 126, Institute of Applied Mathematics, University of Hamburg, Hamburg, Germany, 1997.
9. B. Chen and P.T. Harker, "A Non-Interior-Point Continuation Method for Linear Complementarity Problems," SIAM Journal on Matrix Analysis and Applications, vol. 14, pp. 1168-1190, 1993.
10. B. Chen and P. T. Harker, "Smooth Approximations to Nonlinear Complementarity Problems," SIAM Journal on Optimization, vol. 7, pp. 403-420, 1997.
11. B. Chen and N. Xiu, "A Global Linear and Local Quadratic Non-Interior Continuation Method for Non-linear Complementarity Problems Based on Chen-Mangasarian Smoothing Function," Technical Report, Department of Management and Systems, Washington State University, Pullman, WA, 1997.
12. C. Chen and O.L. Mangasarian, "A Class of Smoothing Functions for Nonlinear and Mixed Complementarity Problems," Computational Optimization and Applications, vol. 5, pp. 97-138, 1996.
13. 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," Mathematics of Computation, vol. 67, pp. 519-540, 1998.
14. X. Chen and Y. Ye, "On Homotopy-Smoothing Methods for Variational Inequalities," SIAM J. Control and Optimization, to appear.
15. R.W. Cottle, J.-S. Pang and R.E. Stone, The Linear Complementarity Problem, Academic Press, Boston, 1992.
16. R.W. Cottle, F. Giannessi and J.-L. Lions, editors, Variational Inequalities and Complementarity Problems: Theory and Applications, J. Wiley: New York, 1980.
17. F. Facchinei and C. Kanzow, "Beyond Monotonicity in Regularization Methods for Nonlinear Complementarity Problems," SIAM J. on Control and Optimization, to appear.
18. M.C. Ferris and J.-S. Pang, editors, "Complementarity and Variational problems: State of the Art, SIAM, Philadelphia, 1997.
19. M.C. Ferris and J.-S. Pang, "Engineering and Economic Applications of Complementarity Problems," SIAM Review, vol. 39, pp. 669-713, 1997.
20. A. Fischer, "A Special Newton-Type Optimization Method," Optimization, vol. 24, pp. 269-284, 1992.
21. S.A. Gabriel and J.J. Moré, "Smoothing of Mixed Complementarity Problems and Variational Problems," Complementarity and Variational Problems: State of the Art, SIAM, Philadelphia, PA, pp. 105-116, 1997.
22. S.A. Gabriel, "A Hybrid Smoothing Method for Mixed Nonlinear Complementarity Problems," Computational Optimization and Applications, to appear.
23. O. Güler, "Existence of Interior Points and Interior Paths in Nonlinear Monotone Complementarity Problems," Mathematics of Operations Research, vol. 18, pp. 128-147, 1993.
24. O. Güler, "Limiting Behavior of the Weighted Central Paths in Linear Programming," Mathematical Programming, vol. 65, pp. 347-363, 1994.
25. M.S. Gowda and R. Sznajder, 'The Generalized Order Complementarity Problem," SIAM Journal on Matrix Analysis and Applications, vol. 15, pp. 779-795, 1994.
26. M.S. Gowda and R. Sznajder, "Weak Univalence and the Connectedness of Inverse Images of Continuous Functions," Research Report 97-02, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland, 1997.
27. P.T. Harker and J.-S. Pang, "Finite Dimension Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms and Applications," Mathematical Programming, vol. 48, pp. 161-220, 1990.
28. K. Hotta and A. Yoshise, "Global Convergence of a Class of Non-Interior-Point Algorithms Using ChenHarker-Kanzow Function for Nonlinear Complementarity Problems," Technical Report 708, Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Ibaraki, Japan, 1996.
29. H. Jiang, "Smoothed Fischer-Burmeister Equation Methods for the Complementarity Problem," Technical Report, Department of Mathematics, the University of Melbourne, Parkville, Victoria, Australia, 1997.
30. H. Jiang and L. Qi, "A New Nonsmooth Equations Approach to Nonlinear Complementarity Problems," SIAM Journal on Control and Optimization, vol. 35, pp. 178-193, 1997.
31. C. Kanzow, "Some Noninterior Continuation Methods for Linear Complementarity Problems," Journal on Matrix Analysis and Applications, vol. 17, pp. 851-868, 1996.
32. C. Kanzow, "A New Approach to Continuation Methods for Complementarity Problems with Uniform P-Functions," Operations Research Letters, vol. 20, pp. 85-92, 1997.
33. M. Kojima, N. Megiddo and T. Noma, "Homotopy Continuation Methods for Nonlinear Complementarity Problems," Mathematics of Operations Research, vol. 16, pp. 754-774, 1991.
34. M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, volume 538 of Lecture Notes in Computer Science, Springer-Verlag: New York, 1991.
35. M. Kojima, M. Mizuno and T. Noma, "A New Continuation Method for Complementarity Problems with Uniform P-Functions," Mathematical Programming, vol. 43, pp. 107-113, 1989.
36. M. Kojima, M. Mizuno and T. Noma, "Limiting Behavior of Trajectories Generated by a Continuation Method for Monotone Complementarity problems," Mathematics of Operations Research, vol. 43, pp. 662-675, 1990.
37. L. McLinden, 'The Complementarity Problem for Maximal Monotone Multifunction," in Variational Inequalities and Complementarity Problems, John Wiley and Sons: New York, pp. 251-270, 1980.
38. N. Megiddo, "Pathways to the Optimal Set in Linear Programming," in Progress in Mathematical Programming: Interior-Point and Related Methods, Springer-Verlag: New York, pp. 131-158, 1989.
39. R.D.C. Monteiro and J.-S. Pang, "7Properties of an Interior-Point Mapping for Mixed Complementarity Problems," Mathematics of Operations Research, vol. 21, pp. 629-654, 1996.
40. J.J. Moré, "Global Methods for Nonlinear Complementarity Problems," Mathematics of Operations Research, vol. 21, pp. 589-614, 1996.
41. Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming, SIAM Publications, 1994.
42. J.-S. Pang, "Complementarity Problems," in Handbook on Global Optimization, Kluwer Academic publishers: Boston, 1994.
43. J.-M. Peng and Z. Lin, "A Non-interior Continuation Method for Generalized Linear Complementarity Problems," Research Report, Academica Sinica, Beijing, China, 1997.
44. G. Di Pillo and F. Giannessi, editors, Nonlinear Optimization and Applications, Plenum Press: New York, 1996.
45. 0-Functions," Research Report, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China, Revised 1997.
46. H.-D. Qi and L.-Z. Liao, "A Smoothing Newton Method for Extended Vertical Linear Complementarity Problems," Research Report, Institute of Computational Mathematics and Scientific Computing, Chinese Academy of Sciences, Beijing, China, 1997.
47. L. Qi and D. Sun, "Globally Linearly, and Globally and Locally Superlinearly Convergent Versions of the Hotta-Yoshise Non-Interior Point Algorithm for Nonlinear Complementarity Problems," Technical Report, School of Mathematics, The University of New South Wales, Sydney, Australia, 1997.
48. L. Qi, D. Sun and G. Zhou, "A New Look at Smoothing Newton Methods for Nonlinear Complementarity Problems and Box Constrained Variational Inequalities," Technical Report, School of Mathematics, The University of New South Wales, Sydney, Australia, 1997.
49. 0-Functions in Box Variational Inequality Problems," Research Report, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD, 1997.
50. P.K. Subramanian, "A Note on Least Two Norm Solutions of Monotone Complementarity Problems," Appl. Math. Letters, vol. 1, pp. 395-397, 1988.
51. D. Sun, "A Regularized Newton Method for Solving Nonlinear Complementarity Problems," Technical Report, School of Mathematics, The University of New South Wales, Sydney, Australia, 1997.
52. 0-Complementarity Problem," in Reformulation-Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers: Boston, to appear.
53. P. Tseng, "Growth Behavior of a Class of Merit Functions for the Nonlinear Complementarity Problem," Journal of Optimization Theory and Applications, vol. 89, pp. 17-37, 1996.
54. P. Tseng, "Analysis of a Non-Interior Continuation Method Based on Chen-Mangasarian Smoothing Functions for Complementarity Problems," Technical Report, Department of Mathematics, University of Washington, Seattle, WA, 1997.
55. S. Xu, "The Global Linear Convergence of an Infeasible Non-Interior Path-Following Algorithm for Complementarity Problems with Uniform P-Functions," Technical Report, University of Washington, Seattle, WA, 1996.
56. 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," Technical Report, University of Washington, Seattle, WA, 1997.
57. S. Xu and J.V Burke, "A Polynomial Time Interior-Point Path-Following Algorithm for LCP Based on Chen-Harker-Kanzow Smoothing Techniques," Technical Report, University of Washington, Seattle, WA, 1997.