A one-parametric class of smoothing functions and an improved regularization Newton method for the NCP

Abstract

In this paper, we introduce a one-parametric class of smoothing functions, which enjoys some favourable properties and includes two famous smoothing functions as special cases. Based on this class of smoothing functions, we propose a regularization Newton method for solving the non-linear complementarity problem. The main feature of the proposed method is that it uses a perturbed Newton equation to obtain the direction. This not only allows our method to have global and local quadratic convergences without strict complementarity conditions, but also makes the regularization parameter converge to zero globally Q-linearly. In addition, we use a new non-monotone line search scheme to obtain the step size. Some numerical results are reported which confirm the good theoretical properties of the proposed method.

Keywords:

• non-linear complementarity problem
• smoothing function
• regularization Newton method
• non-monotone line search

AMS Subject Classifications:

• 90C33
• 65K05

Acknowledgements

The authors are grateful to the referees for their careful reading and helpful suggestions that improved the paper greatly.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partly supported by National Natural Science Foundation of China [grant number 11101248], [grant number 11371306]; Basic and Frontier Technology Research Project of Henan Province [grant number 142300410318], [grant number 142300410437]; and Key Scientific Research Projects of Institutions of Higher Education of Henan Province [grant number 15A110039].