Skip to main content
SpringerLink
Log in

An interior penalty approach to a large-scale discretized obstacle problem with nonlinear constraints

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

We propose an interior penalty method to solve a nonlinear obstacle problem arising from the discretization of an infinite-dimensional optimization problem. An interior penalty equation is proposed to approximate the mixed nonlinear complementarity problem representing the Karush-Kuhn-Tucker conditions of the obstacle problem. We prove that the penalty equation is uniquely solvable and present a convergence analysis for the solution of the penalty equation when the problem is strictly convex. We also propose a Newton’s algorithm for solving the penalty equation. Numerical experiments are performed to demonstrate the convergence and usefulness of the method when it is used for the two non-trivial test problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Bensoussan, A., Lions, J.L.: Applications of Variational Inequalities in Stochastic Control. North-Holland (1978)

  2. Daryina, A.N., Izmailov, A.F., Solodov, M.V.: A class of active-set Newton methods for mixed complementarity problems. SIAM J. Optim. 36, 409–429 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Nielsen, B.F., Skavhaug, O., Tveito, A.: Penalty and front-fixing methods for the numerical solution of American option problems. J. Comput. Financ. 5, 69–97 (2001)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Kanzow, C.: Global optimization techniques for mixed complementarity problems. J. Glob. Optim. 16, 1–21 (2000)

    Article  MathSciNet  Google Scholar 

  8. Li, D., Fukushima, M.: Smoothing Newton and quasi-Newton methods for mixed complementarity problems. Comput. Optim. Appl. 17, 203–230 (2000)

    Article  MathSciNet  Google Scholar 

  9. Potra, F.A., Ye, Y.: Interior-point method for nonlinear complementarity problems. J. Optim. Theory Appl. 88, 617–642 (1996)

    Article  MathSciNet  Google Scholar 

  10. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. I & II, Springer Series in Operations Research. Springer, New York (2003)

    MATH  Google Scholar 

  11. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    Book  Google Scholar 

  12. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  13. Zhao, J.X., Wang, Y.: A full-Newton step non-interior continuation algorithm for a class of complementarity problems. J. Comput. Appl. Math. 10, 2728–2739 (2012)

    Article  MathSciNet  Google Scholar 

  14. Zhao, J.X.: A fixed-point method for a class of super-large scale nonlinear complementarity problem. Comput. Math. Appl. 67, 999–1015 (2014)

    Article  MathSciNet  Google Scholar 

  15. Zhang, K., Wang, S.: Pricing American bond options using a penalty method. Automatica 48, 472–479 (2012)

    Article  MathSciNet  Google Scholar 

  16. Qi, L., Sun, D., Zhou, G.l.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math. Program. Ser. A 87, 1–35 (2000)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Yamashita, N., Fukushima, M.: On stationary points of the implicit Lagrangian for nonlinear complementarity problems. J. Optim. Theory Appl. 84, 653–663 (1995)

    Article  MathSciNet  Google Scholar 

  19. Monteiro, R.D.C., Pang, Jong-Shi: Properties of an interior-point mapping for mixed complementarity problems. Math. Oper. Res. 21, 629–654 (1996)

    Article  MathSciNet  Google Scholar 

  20. 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, 227–254 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Wang, S.: A power penalty method for a finite-dimensional obstacle problem with derivative constraints. Optim. Lett. 8, 1799–1811 (2014)

    Article  MathSciNet  Google Scholar 

  23. Wang, S.: A penalty approach to a discretized double obstacle problem with derivative constraints. J. Glob. Optim. 62, 775–790 (2015)

    Article  MathSciNet  Google Scholar 

  24. Wang, S.: A power penalty method for a bounded nonlinear complementarity problem. Optimization 64, 2377–2394 (2015)

    Article  MathSciNet  Google Scholar 

  25. Wang, S., Zhang, K.: An interior penalty method for a finite-dimensional linear complementarity problem in financial engineering. Optim. Lett. 12, 1161–1178 (2018)

    Article  MathSciNet  Google Scholar 

  26. Wang, S.: An interior penalty method for a large-scale finite-dimensional nonlinear double obstacle problem. Appl. Math. Model. 58, 217–228 (2018)

    Article  MathSciNet  Google Scholar 

  27. Hock, W., Schittkowski, K.: Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems, vol. 187–198. Springer

  28. Li, W., Wang, S.: Penalty approach to the HJB equation arising in European stock option pricing with proportional transaction costs. J. Optim. Theory Appl. 143, 279–293 (2009)

    Article  MathSciNet  Google Scholar 

  29. Li, W., Wang, S.: Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme. J. Indust. Manag. Optim. 9, 365–398 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Li, W., Wang, S.: A numerical method for pricing European options with proportional transaction costs. J. Glob. Optim. 60, 59–78 (2014)

    Article  MathSciNet  Google Scholar 

  31. Zhou, Y.Y., Wang, S., Yang, X.Q.: A penalty approximation method for a semilinear parabolic double obstacle problem. J. Glob. Optim. 60, 531–550 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This research is supported by the China Scholarship Council (CSC) (Grant No. 201706255091) and the National Natural Science Foundation of China(NSFC) (Grant No. 71572125).

Author information

Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300354, People’s Republic of China

    Jian-Xun Zhao

  2. Shenzhen Audencia Business School, WeBank Institute of Fintech, Guangdong Laboratory of Artificial Intelligence and Digital Economics (SZ), Shenzhen University, Shenzhen, 518060, China

    Song Wang

Authors
  1. Jian-Xun Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Song Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian-Xun Zhao.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, JX., Wang, S. An interior penalty approach to a large-scale discretized obstacle problem with nonlinear constraints. Numer Algor 85, 571–589 (2020). https://doi.org/10.1007/s11075-019-00827-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00827-2

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation