zbMATH — the first resource for mathematics

A new one-step smoothing Newton method for nonlinear complementarity problem with \(P_{0}\)-function. (English) Zbl 1239.65034
Summary: A nonlinear complementarity problem with \(P_{0}\)-function is studied. Based on a new smoothing function, the problem is approximated by a family of parameterized smooth equations and we present a new one-step smoothing Newton method to solve it. At each iteration, the proposed method only need to solve one system of linear equations and perform one Armijo-type line search. The algorithm is proved to be convergent globally and superlinearly without requiring strict complementarity at the solution. Numerical experiments demonstrate the feasibility and efficiency of the new algorithm.

65K05 Numerical mathematical programming methods
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
Full Text: DOI
[1] 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 journal on optimization, 9, 605-623, (1999) · Zbl 1037.90052
[2] Chen, B.; Harker, P.T., Smoothing approximations to nonlinear complementarity problems, SIAM journal on optimization, 7, 1, 403-420, (1997) · Zbl 0879.90177
[3] Chen, B.; Chen, X., A global and local superlinear continuation smoothing method for \(P_0\) and \(R_0\) NCP or monotone NCP, SIAM journal on optimization, 9, 3, 624-645, (1999)
[4] Facchinei, F.; Pang, J., Finite-dimensional variational inequalities and complementarity problems, Vol. 1, (2003), Springer-Verlag NewYork, pp. 298-299
[5] Qi, L.; Sun, D.; Zhou, G., A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Mathematical programming, series A, 87, 1-35, (2000) · Zbl 0989.90124
[6] Qi, L.; Sun, D., Improving the convergence of non-interior point algorithm for nonlinear complementarity problems, Mathematics of computation, 69, 283-304, (2000) · Zbl 0947.90117
[7] Zhang, L.; Han, J.; Huang, Z., Superlinear/quadratic one-step smoothing Newton method for \(P_0\)-NCP, Acta math sinica, 26, 2, 117-128, (2005) · Zbl 1124.90037
[8] Huang, Z.H.; Han, J.; Chen, Z., Predictor-corrector smoothing Newton method based on a new smoothing function for solving the nonlinear complementarity, Journal of optimization theory and applications, 117, 1, 39-68, (2003) · Zbl 1044.90081
[9] H. Jiang, Smoothed Fischer-Burmeister equation methods for the complementarity problem, Technical Report, Department of Mathematics, The University of Melbourne, Parville, Victoria, Australia, 1997.
[10] Ferris, M.C.; Pang, J.-S., Engineering and economic applications of complementarity problems, SIAM review, 39, 3, 669-713, (1997) · Zbl 0891.90158
[11] Harker, P.T.; Pang, J.-S., Finite-dimensional variational inequality and non-linear complementarity problems: A survey of theory, algorithms and applications, Mathematical programming, 48, 1, 161-220, (1990) · Zbl 0734.90098
[12] Clark, F.H., Optimization and non-smooth analysis, (1993), JohnWiley and Sons New York
[13] Moré, J.J.; Rheinboldt, W.C., On P- and S-functions and related classes of n-dimensional non-linear mappings, Linear algebra and applications, 6, 1, 45-68, (1973) · Zbl 0247.65038
[14] Qi, L., Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of operations research, 18, 227-244, (1993) · Zbl 0776.65037
[15] Qi, L.; Sun, J., A non-smooth version of newton’s method, Mathematical programming, 58, 353-367, (1993) · Zbl 0780.90090
[16] Mifflin, R., Semismooth and semiconvex functions in constrained optimization, SIAM journal of control optimization, 15, 959-972, (1977) · Zbl 0376.90081
[17] Ma, C.F.; Chen, X.H., The convergence of a one-step smoothing Newton method for \(P_0 - N \text{CP}\) based on a new smoothing NCP-function, Journal of computational and applied mathematics, 216, 1-13, (2008) · Zbl 1140.65046
[18] Ma, C.F.; Chen, L.J.; Wang, D.S., A globally and superlinearly convergent smoothing Broyden-like method for solving nonlinear complementarity problem, Applied mathematics and computation, 198, 592-604, (2008) · Zbl 1140.65045
[19] Chen, X.H.; Ma, C.F., A regularization smoothing Newton method for solving nonlinear complementarity problem, Nonlinear analysis: real world applications, 10, 1702-1711, (2009) · Zbl 1159.90518
[20] Ma, C.F.; Jiang, L.H.; Wang, D.S., The convergence of a smoothing damped gauss – newton method for nonlinear complementarity problem, Nonlinear analysis: real world applications, 10, 2072-2087, (2009) · Zbl 1163.90752
[21] Ma, C.F.; Tang, J., The quadratic convergence of a smoothing levenberg – marquardt method for nonlinear complementarity problem, Applied mathematics and computation, 197, 566-581, (2008) · Zbl 1141.65044
[22] Ma, C.F., A new smoothing quasi – newton method for nonlinear complementarity problems, Applied mathematics and computation, 171, 807-823, (2005) · Zbl 1099.65055
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.