Computational Optimization and Applications

, Volume 34, Issue 2, pp 155–182 | Cite as

Interior-Point Algorithms, Penalty Methods and Equilibrium Problems

  • Hande Y. Benson
  • Arun Sen
  • David F. Shanno
  • Robert J. Vanderbei
Article

Abstract

In this paper we consider the question of solving equilibrium problems—formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPECs)—as nonlinear programs, using an interior-point approach. These problems pose theoretical difficulties for nonlinear solvers, including interior-point methods. We examine the use of penalty methods to get around these difficulties and provide substantial numerical results. We go on to show that penalty methods can resolve some problems that interior-point algorithms encounter in general.

Keywords

interior-point methods nonlinear programming penalty methods equilibrium problems complementarity 

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References

  1. 1.
    M. Anitescu, “Nonlinear programs with unbounded lagrange multiplier sets,” Technical Report ANL/MCS-P793-0200.Google Scholar
  2. 2.
    H.Y. Benson, D.F. Shanno, and R.J. Vanderbei, “Interior point methods for nonconvex nonlinear programming: filter methods and merit functions,” Computational Optimization and Applications, vol. 23, pp. 257–272, 2002.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    H.Y. Benson, D.F. Shanno, and R.J. Vanderbei, “Interior point methods for nonconvex nonlinear programming: Jamming and comparative numerical testing,” Math. Programming, vol. 99, no. 1, pp. 35–48, 2004.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    R.H. Byrd, M.E. Hribar, and J. Nocedal, “An interior point algorithm for large scale nonlinear programming,” SIAM Journal on Optimization, vol. 9, no. 4, pp. 877–900, 1999.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    M.C. Ferris, S.P. Dirkse, and A. Meeraus, “Mathematical programs with equilibrium constraints: Automatic reformulation and solution via constrained optimization,” in Frontiers in Applied General Equilibrium Modeling, T.J. Kehoe, T.N. Srinivasan, and J.Whalley (Eds.), Cambridge University Press, 2005, pp. 67–95.Google Scholar
  6. 6.
    M.C. Ferris and C. Kanzow, “Complementarity and related problems: A survey,” in Handbook of Applied Optimization, P.M. Pardalos and M.G.C. Resende (Eds.), Oxford University Press, New York, 2002, pp. 514–530.Google Scholar
  7. 7.
    M.C. Ferris and J.-S. Pang, “Engineering and economics applications of complementarity problems,” SIAM Review, vol. 39, pp. 669–713, 1997.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    R. Fletcher and S. Leyffer, “Numerical experience with solving MPECs as NLPs,” Optimization Methods and Software, vol. 19, no. 1, pp. 15–40, 2004.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Fletcher, S. Leyffer, and P. L. Toint, “On the global convergence of a filter-SQP algorithm,” SIAM Journal on Optimization, vol. 13, no. 1, pp. 44–59, 2002.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    P.E. Gill, W. Murray, and M.A. Saunders, “snopt: An SQP algorithm for large-scale constrained optimization,” SIAM Journal on Optimization, vol. 12, pp. 979–1006, 2002.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    W. Hock and D. Schittkowski, “Test examples for nonlinear programming codes,” Lecture Notes in Economics and Mathematical Systems 187, Springer Verlag, Heidelberg, 1981.Google Scholar
  12. 12.
    J. Huang and J.-S. Pang, “Option pricing and linear complementarity,” Journal of Computational Finance, vol. 2, no. 3, 1998.Google Scholar
  13. 13.
    S. Leyffer, MacMPEC Test Suite. http://www-unix.mcs.anl.gov/∼leyffer/MacMPEC
  14. 14.
    H. Scheel and S. Scholtes, “Mathematical program with complementarity constraints: Stationarity, optimality and sensitivity,” Mathematics of Operations Research, vol. 25, pp. 1–22, 2000.MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    S. Scholtes, “Convergence properties of a regularization scheme for mathematical programs with complementarity constraints,” SIAM Journal on Optimization, vol. 11, pp. 918–936, 2001.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Sen and D.F. Shanno, “Computing nash equilibria for stochastic games” (working paper).Google Scholar
  17. 17.
    E.M. Simantiraki and D.F. Shanno, “An infeasible interior-point algorithm for solving mixed complementarity problems,” in Complementarity and Variational Problems, State of the Art, Complementarity and Variational Problems, State of the Art, M.C. Ferris and J.S. Pang (Eds.), SIAM, Philadelphia, 1997, pp. 386–404.Google Scholar
  18. 18.
    R.J. Vanderbei and D.F. Shanno, “An Interior-point algorithm for nonconvex nonlinear programming,” Computational Optimization and Applications, vol. 13, pp. 231–252, 1999.MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Waechter and L.T. Biegler, “On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming,” Technical Report RC 23149, IBM T. J. Watson Research Center, Yorktown Heights, NY, March 2004.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Hande Y. Benson
    • 1
  • Arun Sen
    • 2
  • David F. Shanno
    • 3
  • Robert J. Vanderbei
    • 2
  1. 1.Decision Sciences DepartmentLeBow College of Business, Drexel UniversityPhiladelphia
  2. 2.Department of Operations Research and Financial EngineeringPrinceton UniversityPrinceton
  3. 3.RUTCOR - Rutgers Center of Operations ResearchRutgers UniversityNew Brunswick

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