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A box-constrained differentiable penalty method for nonlinear complementarity problems

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An Erratum to this article was published on 17 June 2015

Abstract

In this paper, we propose a box-constrained differentiable penalty method for nonlinear complementarity problems, which not only inherits the same convergence rate as the existing \(\ell _\frac{1}{p}\)-penalty method but also overcomes its disadvantage of non-Lipschitzianness. We introduce the concept of a uniform \(\xi \)–\(P\)-function with \(\xi \in (1,2]\), and apply it to prove that the solution of box-constrained penalized equations converges to that of the original problem at an exponential order. Instead of solving the box-constrained penalized equations directly, we solve a corresponding differentiable least squares problem by using a trust-region Gauss–Newton method. Furthermore, we establish the connection between the local solution of the least squares problem and that of the original problem under mild conditions. We carry out the numerical experiments on the test problems from MCPLIB, and show that the proposed method is efficient and robust.

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  1. http://tresnei.de.unifi.it/.

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Acknowledgments

The authors sincerely thank the two anonymous referees for their careful reading of the paper and their suggestions and questions, all of which greatly improved this paper.

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Authors and Affiliations

  1. Business School of Hunan University, Changsha, 410082, Hunan, People’s Republic of China

    Boshi Tian & Xiaoqi Yang

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Yaohua Hu

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People’s Republic of China

    Xiaoqi Yang

Authors
  1. Boshi Tian
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  2. Yaohua Hu
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  3. Xiaoqi Yang
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Corresponding author

Correspondence to Boshi Tian.

Additional information

Boshi Tian: Research of this author was supported by the NSF (11201383) of China. Xiaoqi Yang: Research of this author was supported by Grants from the Research Grants Council of Hong Kong (PolyU 5292/13E).

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Tian, B., Hu, Y. & Yang, X. A box-constrained differentiable penalty method for nonlinear complementarity problems. J Glob Optim 62, 729–747 (2015). https://doi.org/10.1007/s10898-015-0275-6

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