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A penalty method for a finite-dimensional obstacle problem with derivative constraints

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Abstract

We propose a power penalty method for an obstacle problem arising from the discretization of an infinite-dimensional optimization problem involving differential operators in both its objective function and constraints. In this method we approximate the mixed nonlinear complementarity problem (NCP) arising from the KKT conditions of the discretized problem by a nonlinear penalty equation. We then show the solution to the penalty equation converges exponentially to that of the mixed NCP. Numerical results will be presented to demonstrate the theoretical convergence rates of the method.

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Notes

  1. A more rigorous statement is: find \( u\in H^1_0(0,1)\), where \(H^1_0(0,1)\) denotes the usual Sobolev functional space on \((0,1)\) satisfying \(u(0) = u(1) = 0.\)

References

  1. Angermann, L., Wang, S.: Convergence of a fitted finite volume method for the penalized Black-Scholes equation governing European and American Option pricing. Numer. Math. 106, 1–40 (2007)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Choe, H.J., Shim, Y.: Degenerate variational inequalities with gradient constraints. Annali della Scuola Normale Superiore di Pisa, Classes di Scienze 4e serie, 22, 25–53 (1995)

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

    Google Scholar 

  5. 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 

  6. Davis, M.H.A., Zariphopoulou, T.: American options and transaction fees. In: Davis, M.H.A., et al. (eds.) Mathematical Finance, Springer-Verlag, New York (1995)

  7. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I, II. Springer, New York (2003)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. 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  MATH  MathSciNet  Google Scholar 

  12. 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, 365–389 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. 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  MATH  MathSciNet  Google Scholar 

  14. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. 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  MATH  MathSciNet  Google Scholar 

  17. Wilmott, P., Dewynne, J., Howison, S.: Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford (1993)

    Google Scholar 

Download references

Acknowledgments

Project 11001178 supported by National Natural Science Foundation of China.

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  1. School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Song Wang

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  1. Song Wang
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Correspondence to Song Wang.

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Wang, S. A penalty method for a finite-dimensional obstacle problem with derivative constraints. Optim Lett 8, 1799–1811 (2014). https://doi.org/10.1007/s11590-013-0651-4

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  • DOI: https://doi.org/10.1007/s11590-013-0651-4

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