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A regularized projection method for complementarity problems with non-Lipschitzian functions
Author(s):
Goetz
Alefeld;
Xiaojun
Chen.
Journal:
Math. Comp.
77
(2008),
379-395.
MSC (2000):
Primary 90C33, 65G20
Posted:
June 20, 2007
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Abstract:
We consider complementarity problems involving functions which are not Lipschitz continuous at the origin. Such problems arise from the numerical solution for differential equations with non-Lipschitzian continuity, e.g. reaction and diffusion problems. We propose a regularized projection method to find an approximate solution with an estimation of the error for the non-Lipschitzian complementarity problems. We prove that the projection method globally and linearly converges to a solution of a regularized problem with any regularization parameter. Moreover, we give error bounds for a computed solution of the non-Lipschitzian problem. Numerical examples are presented to demonstrate the efficiency of the method and error bounds.
References:
-
- 1.
- G.E. Alefeld, X. Chen and F.A. Potra, Numerical validation of solutions of complementarity problems: The nonlinear case, Numer. Math., 92 (2002) 1-16. MR 1917363 (2003f:65109)
- 2.
- J.W. Barrett and R.M. Shanahan, Finite element approximation of a model reaction-diffusion problem with a non-Lipschitz nonlinearity, Numer. Math., 59(1991) 217-242. MR 1106381 (92c:65122)
- 3.
- X. Chen and S. Xiang, Computation of error bounds for P-matrix linear complementarity problems, Math. Programming, 106(2006) 513-525. MR 2216793 (2006k:90131)
- 4.
- R.W. Cottle, J.-S. Pang and R.E. Stone, The Linear Complementarity Problem, Academic Press, Boston, MA, 1992. MR 1150683 (93f:90001)
- 5.
- J.E. Dennis and R.E. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, 1983. MR 702023 (85j:65001)
- 6.
- F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems I, II, Springer-Verlag, New York, 2003.
- 7.
- S.A. Gabriel and J.J. Moré, Smoothing of mixed complementarity problems, Complementarity and Variational Problems: State of the Art, eds by M.C. Ferris and J.-S. Pang, (SIAM Publications, Philadelphia, PA, 1997), 105-116. MR 1445075 (98c:90161)
- 8.
- J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 0273810 (42:8686)
- 9.
- J.-S. Pang, Error bounds in mathematical programming, Math. Programming, 79(1997) 299-332. MR 1464772 (98j:90075)
- 10.
- U. Schäfer, An enclosure method for free boundary problems based on a linear complementarity problem with interval data, Numer. Func. Anal. Optim., 22(2001) 991-1011. MR 1871871 (2003d:90110)
- 11.
- P. Tseng, An infeasible path-following method for monotone complementarity problems, SIAM J. Optim., 7(1997) 386-402. MR 1443625 (98c:90169)
- 12.
- R. Varga, Matrix Iterative Analysis, Second Revised and Expanded Edition, Springer-Verlag, New York 2000. MR 1753713 (2001g:65002)
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Additional Information:
Goetz
Alefeld
Affiliation:
Institute of Applied and Numerical Mathematics, University of Karlsruhe (Karlsruhe Institute of Technology KIT), D-76128 Karlsruhe, Germany
Email:
goetz.alefeld@math.uni-karlsruhe.de
Xiaojun
Chen
Affiliation:
Department of Mathematical Sciences, Hirosaki University, 036-8561 Hirosaki, Japan
Email:
chen@cc.hirosaki-u.ac.jp
DOI:
10.1090/S0025-5718-07-02025-X
PII:
S 0025-5718(07)02025-X
Keywords:
Complementarity problems,
non-Lipschitzian continuity,
regularization,
projection,
error bounds.
Received by editor(s):
June 8, 2006
Posted:
June 20, 2007
Copyright of article:
Copyright
2007,
American Mathematical Society
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