On homotopy-smoothing methods for box-constrained variational inequalities

SIAM Journal on Control and Optimization

Volumn 37, Issue 2, 1998, Pages 589-616

  Chen, Xiaojun ^a   Yinyu, Ye ^b  

^a SHIMANE UNIVERSITY   (Japan)

^b UNIVERSITY OF IOWA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; CONVERGENCE OF NUMERICAL METHODS; PROBLEM SOLVING; VARIATIONAL TECHNIQUES;

HOMOTOPY-SMOOTHING METHODS; VARIATIONAL INEQUALITIES;

OPTIMIZATION;

EID: 0345148378     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/s0363012997315907     Document Type: Article

References (42)

1. B.H. AHN, Iterative methods for linear complementarity problem with upperbounds and lowerbounds, Math. Programming, 26 (1983), pp. 265-315.
2. J.V. BURKE AND S. XU, The global linear convergence of a non-interior path-following algorithm for linear complementarity problems, Math. Oper. Res., to appear.
3. B. CHEN, X. CHEN, AND C. KANZOW, A Penalized Fischer-Burmeister NCP-Function: Theoretical Investigation and Numerical Results, Department of Management and Systems, Washington State University, Pullman, WA, 1997.
4. B. CHEN AND P.T. HARKER, A non-interior-point continuation method for linear complementarity problems, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 1168-1190.
5. B. CHEN AND P.T. HARKER, Smooth approximations to nonlinear complementarity problems, SIAM J. Optim., 7 (1997), pp. 403-420.
6. C. CHEN AND O.L. MANGASARIAN, A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 5 (1996), pp. 97-138.
7. X. CHEN, Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equations, J. Comput. Appl. Math., 80 (1997), pp. 105-126.
8. X. CHEN AND L. QI, A parameterized Newton method and a Broyden-like method for solving nonsmooth equations, Comput. Optim. Appl., 3 (1994), pp. 157-179.
9. 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. Comp., 67 (1998), pp. 519-540.
10. X. CHEN AND T. YAMAMOTO, Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Punct. Anal. Optim., 10 (1989), pp. 37-48.
11. S.P. DIRKSE AND M.C. FERRIS, The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems, Optim. Methods Softw., 5 (1995), pp. 123-156.
12. B.C. EAVES, On the basic theorem of complementarity, Math. Programming, 1 (1971), pp. 68-75.
13. Y. FATHI, Computational complexity of LCPs associated with positive definite matrices, Math. Programming, 17 (1979), pp. 335-344.
14. 0 nonlinear complementarity problems, Math. Oper. Res, to appear.
15. 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, PA, 1997, pp. 76-90.
16. A. FISCHER AND C. KANZOW, On the finite termination of an iterative method for linear complementarity problems, Math. Programming, 74 (1996), pp. 279-292.
17. M.C. FERRIS AND J.S. PANG, Engineering and economic applications of complementarity problems, SIAM Rev., 39 (1997), pp. 669-713.
18. S.A. GABRIEL, A hybrid smoothing method for mixed nonlinear complementarity problems, Comput. Optim. Appl., 9 (1998), pp. 153-173.
19. S.A. GABRIEL AND J.J. MORÉ, Smoothing of mixed complementarity problems, in Complementarity and Variational Problems: State of the Art, M.C. Ferris and J.S. Pang, eds., SIAM, Philadelphia, PA, 1997, pp. 105-116.
20. P.T. HARKER AND J.-S. PANG, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programming, 48 (1990), pp. 161-220.
21. P.T. HARKER AND J.-S. PANG, A damped Newton method for the linear complementarity problem, in Computational Solution of Nonlinear Systems of Equations, Lectures in Appl. Math. 26, E.L. Allgower and K. George, eds., AMS, Providence, RI, 1990, pp. 265-284.
22. 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 708, Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Ibaraki 305, Japan, December 1996.
23. C. KANZOW, Some noninterior continuation methods for linear complementarity problems, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 851-868.
24. C. KANZOW AND M. FUKUSHIMA, Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities, Math. Programming, to appear.
25. C. KANZOW, N. YAMASHITA, AND M. FUKUSHIMA, New NCP-functions and their properties, J. Optim. Theory Appl., 94 (1997), pp. 115-135.
26. C.T. KELLEY, Identification of the support of nonsmoothness, in Large Scale Optimization: State of the Art, W.W. Hager, D.W. Hearn, and P.M. Pardalos, eds., Kluwer Academic Publishers B.V., Boston, 1993, pp. 192-205.
27. C.T. KELLEY AND E.W. SACHS, Multilevel algorithms for constrained compact fixed point problems, SIAM J. Sci. Comput., 15 (1994), pp. 645-667.
28. M. KOJIMA, N. MEGIDDO, AND T. NOMA, Homotopy continuation methods for nonlinear complementarity problems, Math. Oper. Res., 16 (1991), pp. 754-774.
29. M. KOJIMA, N. MEGIDDO, T. NOMA, AND A. YOSHISE, A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Comput. Sci. 538, Springer-Verlag, New York, 1991.
30. M. KOJIMA, N. MEGIDDO, AND Y. YE, An interior point potential reduction algorithm for the linear complementarity problem, Math. Programming, 54 (1992), pp. 267-279.
31. 1 equations, J. Oper. Res. Soc. Japan, 29 (1986), pp. 352-374.
32. K.G. MURTY, Linear Complementarity, Linear and Nonlinear Programming, Sigma Ser. Appl. Math. 3, Heldermann-Verlag, Berlin, 1988.
33. J.M. ORTEGA AND W.C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
34. F. POTRA AND Y. YE, Interior-point methods for nonlinear complementarity problems, J. Optim. Theory Appl., 88 (1996), pp. 617-642.
35. L. QI, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18 (1993), pp. 227-244.
36. L. QI AND X. CHEN, A globally convergent successive approximation method for severely nonsmooth equations, SIAM J. Control Optim., 33 (1995), pp. 402-418.
37. 0-Functions in Box Variational Inequality Problems, Tech. report, Department of Mathematics and Statistics, University of Maryland, Baltimore, MD, 1997.
38. S.M. ROBINSON, Newton's method for a class of nonsmooth functions, Set-Valued Anal., 2 (1994), pp. 291-305.
39. P. TSENG, Simplified analysis of an o(nL)-iteration infeasible predictor-corrector path-following method for monotone LCP, in Recent Trends in Optimization Theory and Applications, R.P. Agarwal, ed., World Scientific Press, Singapore, 1994, pp. 423-434.
40. S. XU AND J.V. BURKE, A Polynomial Time Interior-Point Path-Following Algorithm for LCP Based on Chen-Harker-Kanzow Smoothing Techniques, preprint. Department of Mathematics, University of Washington, Seattle, WA, 1996.
41. Y. YE AND K. ANSTREICHER, On quadratic and O(√NL) convergence of a predictor-corrector algorithm for LCP, Math. Programming, 62 (1993), pp. 537-552.
42. Y. YE, On the finite convergence of interior-point algorithms for linear programming, Math. Programming, 57 (1992), pp. 325-335.