Interior-Point Algorithms, Penalty Methods and Equilibrium Problems
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.
Keywordsinterior-point methods nonlinear programming penalty methods equilibrium problems complementarity
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- 1.M. Anitescu, “Nonlinear programs with unbounded lagrange multiplier sets,” Technical Report ANL/MCS-P793-0200.Google Scholar
- 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.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
- 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.J. Huang and J.-S. Pang, “Option pricing and linear complementarity,” Journal of Computational Finance, vol. 2, no. 3, 1998.Google Scholar
- 13.S. Leyffer, MacMPEC Test Suite. http://www-unix.mcs.anl.gov/∼leyffer/MacMPEC
- 16.A. Sen and D.F. Shanno, “Computing nash equilibria for stochastic games” (working paper).Google Scholar
- 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
- 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