A variant smoothing Newton method for \(P_0\)-\(NCP\) based on a new smoothing function.

*(English)*Zbl 1163.65043Recently, there have been strong interests in smoothing Newton methods for solving the non-linear complementarity problems. The idea of smoothing Newton method is to use a smooth function to reformulate the problem concerned as a family of parameterized smooth equations to solve the smooth equations approximately by using the Newton method per iteration. By reducing the parameter to zero, it is hoped that a solution of the original problem can be found.

In this paper, the authors present a new one-step smoothing Newton method proposed for solving the nonlinear complementarity problem (NCP) with \(P_0\)-function based on a new smoothing \(NCP\)-function. They adopt a variant merit function. This algorithm needs only to solve one system of linear equations and performs one line search per iteration. It shows that any accumulation point of the iteration sequence generated by the algorithm is a solution of \(P_0\)-\(NCP\). Furthermore, under the assumption that the solution set is non-empty and bounded, at least one accumulation point of the generated sequence can be guaranteed. Numerical experiments show the feasibility and efficiency of the algorithm.

In this paper, the authors present a new one-step smoothing Newton method proposed for solving the nonlinear complementarity problem (NCP) with \(P_0\)-function based on a new smoothing \(NCP\)-function. They adopt a variant merit function. This algorithm needs only to solve one system of linear equations and performs one line search per iteration. It shows that any accumulation point of the iteration sequence generated by the algorithm is a solution of \(P_0\)-\(NCP\). Furthermore, under the assumption that the solution set is non-empty and bounded, at least one accumulation point of the generated sequence can be guaranteed. Numerical experiments show the feasibility and efficiency of the algorithm.

Reviewer: Nada Djuranović-Miličić (Belgrade)

##### MSC:

65K05 | Numerical mathematical programming methods |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

##### Keywords:

\(NCP\); smoothing Newton method; non-linear complementarity problems; algorithm; numerical experiments
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\textit{J. Zhang} and \textit{K.-C. Zhang}, J. Comput. Appl. Math. 225, No. 1, 1--8 (2009; Zbl 1163.65043)

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##### References:

[1] | 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 |

[2] | Ferris, M.C.; Pang, J.-S., Engineering and economic applications of complementarity problems, SIAM review, 39, 3, 669-713, (1997) · Zbl 0891.90158 |

[3] | Facchinei, F.; Kanzow, C., Beyond monotonicity in regularization methods for non-linear complementarity problems, SIAM journal on control and optimization, 37, 2, 1150-1161, (1999) · Zbl 0997.90085 |

[4] | Chen, B.; Harker, P.T., A non-interior-point continuation method for linear complementarity problems, SIAM journal on matrix analysis and applications, 14, 2, 1168-1190, (1993) · Zbl 0788.65073 |

[5] | Chen, B.; Harker, P.T., Smoothing approximations to nonlinear complementarity problems, SIAM journal on optimization, 7, 1, 403-420, (1997) · Zbl 0879.90177 |

[6] | 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, 2, 605-623, (1999) · Zbl 1037.90052 |

[7] | Chen, B.; Chen, X., A global and local superlinear continuation smoothing method for \(P_0\) and \(R_0 N C P\) or monotone \(N C P\), SIAM journal on optimization, 9, 3, 624-645, (1999) |

[8] | Chen, X.; Qi, L.; Sun, D., Global and superlinear convergence of the smoothing Newton method and its application to general box-constrained variational inequalities, Mathematics of computation, 67, 1, 519-540, (1998) · Zbl 0894.90143 |

[9] | Qi, H., A regularized smoothing Newton method for box constrained variational inequality problems with \(P_0\)-functions, SIAM journal on optimization, 10, 1, 315-330, (2000) · Zbl 0955.90136 |

[10] | Qi, L.; Sun, D., Improving the convergence of non-interior point algorithm for nonlinear complementarity problems, Mathematics of computation, 69, 1, 283-304, (2000) · Zbl 0947.90117 |

[11] | Tseng, P., Error bounds and superlinear convergence analysis of some Newton-type methods in optimization, (), 445-462 · Zbl 0965.65091 |

[12] | Zhang, L.; Han, J.; Huang, Z., Superlinear/quadratic one-step smoothing Newton method for \(P_0 - N C P\), Acta math sinica, 26, 2, 117-128, (2005) · Zbl 1124.90037 |

[13] | Huang, Z.H.; Han, J.; Chen, Z., Predictor-corrector smoothing Newton method based on a new smoothing function for solving the nonlinear complementarity problem with a \(P_0\) function, Journal of optimization theory and applications, 117, 1, 39-68, (2003) · Zbl 1044.90081 |

[14] | Ma, C.-F.; Chen, X., The convergence of a one-step smoothing Newton method for \(P_0 - N C P\) based on a new smoothing NCP-function, Journal of computational and applied mathematics, 216, 1, 1-13, (2008) · Zbl 1140.65046 |

[15] | 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 |

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