Parallel Newton methods for the nonlinear complementarity problem
Article
Received:
Revised:
- 57 Downloads
- 5 Citations
Abstract
In this paper, we discuss how the basic Newton method for solving the nonlinear complementarity problem can be implemented in a parallel computation environment. We propose some synchronized and asynchronous Newton methods and establish their convergence.
Key words
Nonlinear complementarity problem Newton method parallel computing synchronized and asynchronous methodsPreview
Unable to display preview. Download preview PDF.
References
- [1]G.M. Baudet, “Asynchronous iterative methods for multiprocessors,”Journal of the ACM 2 (1978) 226–244.Google Scholar
- [2]A. Berman and R.J. Plemmons,Nonnegative Matrices in the Mathematical Sciences (Academic Press, New York, 1979).Google Scholar
- [3]D.P. Bertsekas, “Distributed asynchronous computation of fixed points,”Mathematical Programming 27 (1983) 107–120.Google Scholar
- [4]B.C. Eaves, “Where solving for stationary points by LCPs is mixing Newton iterates,” in B.C. Eaves, F.J. Gould, H.O. Peitgen and M.J. Todd eds.,Homotopy Methods and Global Convergence (Plenum Press, New York, 1983) pp. 63–78.Google Scholar
- [5]N.H. Josephy, “Newton's method for generalized equations,” Technical Summary Report No. 1965, Mathematics Research Center, University of Wisconsin (Madison, Wisconsin, 1979).Google Scholar
- [6]N.H. Josephy, “Quasi-Newton methods for generalized equations,” Technical Summary Report No. 1966, Mathematics Research Center, University of Wisconsin (Madison, Wisconsin, 1979).Google Scholar
- [7]L. Kronsjo, Computational Complexity of Sequential and Parallel Algorithms (John Wiley and Sons, Great Britain, 1985).Google Scholar
- [8]H.T. Kung, “Synchronized and asynchronous parallel algorithms for multiprocessors,” in J.F. Traub, ed.,Algorithms and Complexity: New Directions and Recent Results (Academic Press, New York, 1976) 153–200.Google Scholar
- [9]O.L. Mangasarian and R. De Leone, “Parallel successive overrelaxation methods for symmetric linear complementarity problems and linear programs,” Computer Sciences Technical Report #647, University of Wisconsin (Madison, Wisconsin, 1986).Google Scholar
- [10]O.L. Mangasarian and R. De Leone, “Parallel gradient projection successive overrelaxation methods for symmetric linear complementarity problems and linear programs,” Computer Sciences Technical Report #659, University of Wisconsin (Madison, Wisconsin, 1986).Google Scholar
- [11]H. Mukai, “Parallel algorithms for solving systems of nonlinear equations,”Proceedings of the 17th Annual Allerton Conference on Communications, Control and Computations (October 10–12, 1979) pp. 37–46.Google Scholar
- [12]J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).Google Scholar
- [13]J.S. Pang, “On the convergence of a basic iterative scheme for the implicit complementarity problem,”Journal of Optimization Theory and Applications 37 (1982) 149–162.Google Scholar
- [14]J.S. Pang, “Necessary and Sufficient Conditions for the convergence of iterative methods for the linear complementarity problem,”Journal of Optimization Theory and Applications 42 (1984) 1–18.Google Scholar
- [15]J.S. Pang, “Inexact Newton methods for the nonlinear complementarity problem,”Mathematical Programming 36 (1986) 54–71.Google Scholar
- [16]J.S. Pang and D. Chan, “Iterative methods for variational and complementarity problems,”Mathematical Programming 24 (1982) 284–313.Google Scholar
- [17]J.S. Pang and J.M. Yang, “Two-stage parallel iterative methods for the symmetric linear complementarity problem,” to appear inAnnals of Operations Research (1988).Google Scholar
- [18]S.M. Robinson, “Strongly Regular Generalized Equations,”Mathematics of Operations Research 5 (1980) 43–62.Google Scholar
- [19]V.E. Shamanskii, “On a modification of the Newton method,”Ukrainskiî Matematiceskiî Žurnal 19 (1967) 133–138.Google Scholar
- [20]K.M. Thompson, “Parallel algorithms for solving the linear complementarity problem,” paper presented at the TIMS/ORSA meeting (New Orleans, May 4–6, 1987).Google Scholar
- [21]K.M. Thompson, “A parallel asynchronous successive overrelaxation algorithm for solving the linear complementarity problem,” Computer Sciences Technical Report #705, University of Wisconsin (Madison, Wisconsin, 1987).Google Scholar
- [22]J.F. Traub,Iterative Methods for the Solution of Equations (Prentice-Hall, New Jersey, 1964).Google Scholar
Copyright information
© The Mathematical Programming Society, Inc. 1988