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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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Notes

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

References

  1. Bensoussan, A., Lions, J.L.: Applications of Variational Inequalities in Stochastic Control. North Holland, New York (1982)

    MATH  Google Scholar 

  2. Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  3. Billups, S.C., Dirkse, S.P., Ferris, M.C.: A comparison of large scale mixed complementarity problem solvers. Comput. Optim. Appl. 7(1), 3–25 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, B., Harker, P.T.: Smooth approximations to nonlinear complementarity problems. SIAM J. Optim. 7(2), 403–420 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5(2), 97–138 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust Region Methods. SIAM, Philadelphia (1987)

    Google Scholar 

  7. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  8. De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75(3), 407–439 (1996)

    Article  MATH  Google Scholar 

  9. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, Berlin (2003)

    Google Scholar 

  11. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. II. Springer, Berlin (2003)

    Google Scholar 

  12. Fackler, P.L.: Applied Computational Economics and Finance. The MIT Press, Cambridge (2002)

    MATH  Google Scholar 

  13. Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39(4), 669–713 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fischer, A.: A special Newton-type optimization method. Optimization 24(3–4), 269–284 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fletcher, R.: Practical Methods of Optimization. Wiley, New York (2013)

    Google Scholar 

  16. Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48(1–3), 161–220 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang, C.C., Wang, S.: A power penalty approach to a nonlinear complementarity problem. Oper. Res. Lett. 38(1), 72–76 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huang, C.C., Wang, S.: A penalty method for a mixed nonlinear complementarity problem. Nonlinear Anal. Theory Methods Appl. 75(2), 588–597 (2012)

    Article  MATH  Google Scholar 

  19. Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28(3), 533–552 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jiang, H.Y., Qi, L.Q.: A new nonsmooth equations approach to nonlinear complementarity problems. SIAM J. Control Optim. 35(1), 178–193 (1997)

    Article  MathSciNet  Google Scholar 

  21. Kanzow, C., Yamashita, N., Fukushima, M.: New NCP-functions and their properties. J. Optim. Theory Appl. 94(1), 115–135 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kanzow, C., Yamashita, N., Fukushima, M.: Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints. J. Comput. Appl. Math. 173(2), 321–343 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Konnov, I.V.: Properties of gap functions for mixed variational inequalities. Sib. J. Numer. Math. 3(3), 259–270 (2000)

    MATH  Google Scholar 

  24. Macconi, M., Morini, B., Porcelli, M.: Trust-region quadratic methods for nonlinear systems of mixed equalities and inequalities. Appl. Numer. Math. 59(5), 859–876 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Moré, J., Rheinboldt, W.: On P- and S-functions and related classes of \(n\)-dimensional nonlinear mappings. Linear Algebra Appl. 6, 45–68 (1973)

    Article  MATH  Google Scholar 

  26. Morini, B., Porcelli, M.: TRESNEI, a Matlab trust-region solver for systems of nonlinear equalities and inequalities. Comput. Optim. Appl. 51(1), 27–49 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (2006)

    MATH  Google Scholar 

  28. Qi, L.Q., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58(1), 353–367 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rubinov, A.M., Yang, X.Q.: Lagrange-Type Functions in Constrained Non-convex Optimization. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  30. Wang, S., Yang, X.Q.: A power penalty method for linear complementarity problems. Oper. Res. Lett. 36(2), 211–214 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, S., Yang, X.Q., Teo, K.L.: Power penalty method for a linear complementarity problem arising from American option valuation. J. Optim. Theory Appl. 129(2), 227–254 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, X.Q., Huang, X.X.: A nonlinear Lagrangian approach to constrained optimization problems. SIAM J. Optim. 11(4), 1119–1144 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yu, C.J., Teo, K.L., Zhang, L.S., Bai, Y.Q.: A new exact penalty function method for continuous inequality constrained optimization problems. J. Ind. Manag. Optim. 6(4), 895 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zang, I.: A smoothing-out technique for min–max optimization. Math. Program. 19(1), 61–77 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, K.: American Option Pricing and Penalty Methods. Ph.D. thesis, The Hong Kong Polytechnic University (2006)

  36. Zvan, R., Forsyth, P.A., Vetzal, K.R.: Penalty methods for American options with stochastic volatility. J. Comput. Appl. Math. 91(2), 199–218 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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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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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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  • DOI: https://doi.org/10.1007/s10898-015-0275-6

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