Skip to main content
SpringerLink
Log in

A penalty approach to a discretized double obstacle problem with derivative constraints

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

This work presents a penalty approach to a nonlinear optimization problem with linear box constraints arising from the discretization of an infinite-dimensional differential obstacle problem with bound constraints on derivatives. In this approach, we first propose a penalty equation approximating the mixed nonlinear complementarity problem representing the Karush–Kuhn–Tucker conditions of the optimization problem. We then show that the solution to the penalty equation converges to that of the complementarity problem with an exponential convergence rate depending on the parameters used in the equation. Numerical experiments, carried out on a non-trivial test problem to verify the theoretical finding, show that the computed rates of convergence match the theoretical ones well.

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

Similar content being viewed by others

Notes

  1. For simplicity, we omit the introduction of Sobolev function spaces.

References

  1. Azevedo, A., Miranda, F., Santos, L.: Variational and quasivariational inequalities with first order constraints. J. Math. Anal. Appl. 397, 738–756 (2013)

    Article  MathSciNet  Google Scholar 

  2. Barrett, J.W., Prigozhin, L.: A quasi-variational inequality problem in superconductivity. Math. Models Methods Appl. Sci. 20, 679–706 (2010)

    Article  MathSciNet  Google Scholar 

  3. Cardaliaguet, P.: A double obstacle problem arising in differential game theory. J. Math. Anal. Appl. 360, 95–107 (2009)

    Article  MathSciNet  Google Scholar 

  4. 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 

  5. Chen, M., Huang, C.: A power penalty method for the general traffic assignment problem with elastic demand. J. Ind. Manag. Optim. 10, 1019–1030 (2014)

    Article  MathSciNet  Google Scholar 

  6. Choe, H.J., Shim, Y.: Degenerate variational inequalities with gradient constraints. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 4(22), 25–53 (1995)

    MathSciNet  Google Scholar 

  7. Damgaard, A.: Computation of reservation prices of options with proportional transaction costs. J. Econ. Dyn. Control 30, 415–444 (2006)

    Article  MathSciNet  Google Scholar 

  8. 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  Google Scholar 

  9. Davis, M.H.A., Zariphopoulou, T.: American options and transaction fees. In: Davis, M.H.A., et al. (eds.) Mathematical Finance. Springer, Berlin (1995)

    Chapter  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).

  11. Hoang, N.S., Ramm, A.G.: An iterative scheme for solving nonlinear equations with monotone operators. BIT 48, 725–741 (2008)

    Article  MathSciNet  Google Scholar 

  12. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin, Heidelberg (1996)

    Book  Google Scholar 

  13. 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 

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

    Article  MathSciNet  Google Scholar 

  15. Huang, C.S., Wang, S., Teo, K.L.: On application of an alternating direction method to Hamilton–Jacobin–Bellman equations. J. Comput. Appl. Math. 166, 153–166 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Kunze, M., Rodrigues, J.F.: An elliptic quasi-variational inequality with gradient constraints and some of its applications. Math. Methods Appl. Sci. 23, 897–908 (2000)

    Article  MathSciNet  Google Scholar 

  18. Lesmana, D.C., Wang, S.: An upwind finite difference method for a nonlinear BlackScholes equation governing European option valuation under transaction costs. Appl. Math. Comput. 219, 8811–8828 (2013)

    Article  MathSciNet  Google Scholar 

  19. 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 

  20. 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 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  23. Mangasarian, O.L., Solodov, M.V.: Nonlinear complementarity as unconstrained and constrained minimization. Math. Program. 62, 277–297 (1993)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Santos, L.: Variational problems with non-constant gradient constraints. Port. Math. (N.S.) 59, 205–248 (2002)

    MathSciNet  Google Scholar 

  26. Wang, S.: A novel fitted finite volume method for the Black-Scholes equation governing option pricing. IMA J. Numer. Anal. 24, 699–720 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. 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 

  29. 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 

  30. 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 

  31. Zakamouline, V.: European option pricing and hedgeing with both fixed and proportional transaction costs. J. Econ. Dyn. Control 30, 1–25 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  33. Zhou, Y.Y., Wang, S., Yang, X.Q.: A penalty approximation method for a semilinear parabolic double obstacle problem, J. Glob. Optim. in press, doi:10.1007/s10898-013-0122-6

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, WA, 6845, Australia

    Song Wang

Authors
  1. Song Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Song Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S. A penalty approach to a discretized double obstacle problem with derivative constraints. J Glob Optim 62, 775–790 (2015). https://doi.org/10.1007/s10898-014-0262-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-014-0262-3

Keywords

Search

Navigation