zbMATH — the first resource for mathematics

The non-interior continuation methods for solving the \(P_0\) function nonlinear complementarity problem. (English) Zbl 1002.90072
Summary: We propose a new smoothing function that possesses a property not satisfied by the existing smoothing functions. Based on this smoothing function, we discuss the existence and continuity of the smoothing path for solving the \(P_0\) function Nonlinear Complementarity Problem (NCP). Using the characteristics of the new smoothing function, we investigate the boundedness of the iteration sequence generated by the non-interior continuation methods for solving the \(P_0\) function NCP under the assumption that the solution set of the NCP is nonempty and bounded. We show that the assumption that the solution set of the NCP is nonempty and bounded is weaker than those required by other continuation methods for solving the NCP.

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
Full Text: DOI
[1] Burke, J., Xu, S., The global linear convergence of a non-interior path-following algorithm for linear complementarity problems, Math. Oper. Res., 1998, 23(3): 719–734. · Zbl 0977.90056
[2] Chen, B., Chen, X., A global and local superlinear continuation-smoothing method for P0 + R{IN0} and monotone NCP, SIAM J. Optim., 1999, 9(3): 624–645. · Zbl 0955.90132
[3] Chen, B., Xiu, N., A global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions, SIAM J. Optim., 1999, 9(3): 605–623. · Zbl 1037.90052
[4] Facchinei, F., Kanzow, C., Beyond monotonicity in regularization methods for nonlinear complementarity problems, SIAM J. Control Optim., 1999, 37(4): 1150–1161. · Zbl 0997.90085
[5] Gowda, M. S., Tawhid, M. A., Existence and limiting behavior of trajectories associated with P0 - equations, Comput. Optim. Appl., 1999, 12(1/2/3): 229–251. · Zbl 1040.90563
[6] Sun, D., A regularization Newton method for solving nonlinear complementarity problems, Appl. Math. Optim., 1999, 40 (3): 315–339. · Zbl 0937.90110
[7] Ravindran, G., Gowda, M.S., Regularization of P0 - functions in box variational inequality problems, SIAM J. Optim., 2000, 11: 748–760. · Zbl 1010.90083
[8] Chen, B., Harker. P. T., A non-interior-point continuation method for linear complementarity problem, SIAM J. Matrix Anal. Appl., 1993, 14(4): 1168–1190. · Zbl 0788.65073
[9] Billus, S. C., Dirkse, S. P., Fems, M. C., A comparison of algorithms for large-scale mixed complementarity problems, Comput. Optim. Appl., 1997, 7(1): 3–25. · Zbl 0883.90116
[10] Kanzow, C., Some noninterior continuation methods for linear complementarity problems, SIAM J. Matrix Anal. Appl., 1996, 17(4): 851–868. · Zbl 0868.90123
[11] Smale, S., Algorithms for solving equations, in Proceeding of International Congress of Mathematicians ( ed. Gleason, A. M.), Providence, Rhode Island: American Mathematics Society, 1987, 172–195. · Zbl 0665.65058
[12] Chen, X., Qi, L., A parameterized Newton method and a Broyden-like method for solving nonsmooth equations, Comput. Optim. Appl., 1995, 5: 97–138.
[13] Chen, C., Mangasarian, O. L., A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 1996, 5: 97–138. · Zbl 0859.90112
[14] Tseng, P., Analysis of a non-interior continuation method based on Chen-Mangasarian smoothing functions for complementarity problems, Reformulation-Nonsmootb, PiecewiseSmooth, Semismooth and Smoothing Methods (eds. Fukushima, M., Qi, L.), Boston: Kluwer Academic Publishers, 1999, 381–404. · Zbl 0928.65078
[15] Gabriel, S. A., MonĂ©, J. J., Smoothing of mixed complementarity problems. Complementarity and Variational Problems: State of the Art (eds. Fems, M. C., Pang, J. S.), Philadelphia, Pennsylvania: SIAM Publishing, 1997, 105–116. · Zbl 0886.90154
[16] Fischer, A., A special Newton-type optimization method, Optim., 1992, 24: 269–284. · Zbl 0814.65063
[17] Qi, H. D., A regularization smoothing Newton method for box constrained variational inequality pmblems with P0 - functions, SIAM J. Optim., 1999, 10(2): 315–330. · Zbl 0955.90136
[18] Kojima, M., Megiddo, N., Noma, T. et al., A unified approach to intrior point algorithm for linear complementarity problems, Lecture Notes in Computer Science, Berlin: Springer-Verlag, 1991, Vol. 538. · Zbl 0745.90069
[19] Palais, R. S., Temg, C. L., Critical Point Theory Submanifold Geometry, Lecture Notes in Mathematics, Vol. 1353, Berlin: Springer-Verlag, 1988.
[20] Isac, G., Bulavski, V., Kalashnikov, V., Exceptional families, topological degree and complementarity pmblems, J. Global Optim., 1997, 10(2): 207–225. · Zbl 0880.90127
[21] Zhao, Y., Han, J., Qi, H., Exceptional families and existence theorems for variational inequality pmblems, J. Optim. Theory Appl., 1999, 101(2): 475–495. · Zbl 0947.49005
[22] Kanzow, C., A new approach to continuation methods for complementarity problem with uniform P- function, Oper. Res. Letters, 1997, 20(2): 85–92. · Zbl 0890.90169
[23] Chen, B., Harker, P. T., Pmar, M. C., Continuation method for nonlinear complementarity problems via normal maps, European J. Oper. Res., 1999, 116: 591–606. · Zbl 1009.90118
[24] Chen, B., Harker, P. T., Smooth approximation to nonlinear complementarity pmblems, SIAM J. Optim., 1997, 7(2): 403–420. · Zbl 0879.90177
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.