Solving equilibrium problems using extended mathematical programming

  • Youngdae KimEmail author
  • Michael C. Ferris
Full Length Paper


We introduce an extended mathematical programming framework for specifying equilibrium problems and their variational representations, such as generalized Nash equilibrium, multiple optimization problems with equilibrium constraints, and (quasi-) variational inequalities, and computing solutions of them from modeling languages. We define a new set of constructs with which users annotate variables and equations of the model to describe equilibrium and variational problems. Our constructs enable a natural translation of the model from one formulation to another more computationally tractable form without requiring the modeler to supply derivatives. In the context of many independent agents in the equilibrium, we facilitate expression of sophisticated structures such as shared constraints and additional constraints on their solutions. We define shared variables and demonstrate their uses for sparse reformulation, economic equilibrium problems sharing economic states, mixed pricing behavior of agents, and so on. We give some equilibrium and variational examples from the literature and describe how to formulate them using our framework. Experimental results comparing performance of various complementarity formulations for shared variables are provided. Our framework has been implemented and is available within GAMS/EMP.


Equilibrium programming Nash equilibrium problems Quasi-variational inequalities 

Mathematics Subject Classification

90C33 90C90 65K10 65K15 



This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contract number DE-AC02-06CH11357.


  1. 1.
    Aguiar, A., Narayanan, B., Mcdougall, R.: An overview of the GTAP 9 data base. J. Glob. Econ. Anal. 1(1), 181–208 (2016)Google Scholar
  2. 2.
    Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Britz, W., Ferris, M., Kuhn, A.: Modeling water allocating institutions based on multiple optimization problems with equilibrium constraints. Environ. Model. Softw. 46, 196–207 (2013)Google Scholar
  4. 4.
    Brook, A., Kendrick, D., Meeraus, A.: GAMS: A User’s Guide. The Scientific Press, South San Francisco (1988)Google Scholar
  5. 5.
    Davis, T.A.: UMFPACK (2007). Accessed 14 Dec 2017
  6. 6.
    Dirkse, S.P., Ferris, M.C.: The PATH solver: a non-monotone stabilization scheme for mixed complementarity problems. Optim. Methods Softw. 5(2), 123–156 (1995)Google Scholar
  7. 7.
    An EMP framework for equilibrium problems (2019). Accessed 16 Feb 2019
  8. 8.
    Facchinei, F., Fischer, A., Piccialli, V.: On generalized Nash games and variational inequalities. Oper. Res. Lett. 35(2), 159–164 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Annals of Operations Research 175(1), 177–211 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ferris, M.C., Dirkse, S.P., Jagla, J.H., Meeraus, A.: An extended mathematical programming framework. Comput. Chem. Eng. 33(12), 1973–1982 (2009)Google Scholar
  11. 11.
    Ferris, M.C., Fourer, R., Gay, D.M.: Expressing complementarity problems in an algebraic modeling language and communicating them to solvers. SIAM J. Optim. 9(4), 991–1009 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ferris, M.C., Munson, T.S.: Interfaces to PATH 3.0: design, implementation and usage. Comput. Optim. Appl. 12(1), 207–227 (1999)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming, 2nd edn. Cengage Learning, Boston (2002)zbMATHGoogle Scholar
  14. 14.
    Harker, P.T.: A variational inequality approach for the determination of oligopolistic market equilibrium. Math. Program. 30, 105–111 (1984)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Harker, P.T.: Multiple equilibrium behaviors on networks. Transp. Sci. 22(1), 39–46 (1988)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Harker, P.T.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)zbMATHGoogle Scholar
  17. 17.
    Haurie, A., Krawczyk, J.B.: Optimal charges on river effluent from lumped and distributed sources. Environ. Model. Assess. 2(3), 177–189 (1997)Google Scholar
  18. 18.
    Kim, Y., Ferris, M.C.: SELKIE: a model transformation and distributed solver for structured equilibrium problems. Technical Report, University of Wisconsin-Madison, Department of Computer Sciences (2017)Google Scholar
  19. 19.
    Krawczyk, J.B., Uryasev, S.: Relaxation algorithms to find Nash equilibria with economic applications. Environ. Model. Assess. 5(1), 63–73 (2000)Google Scholar
  20. 20.
    Leyffer, S., Munson, T.: Solving multi-leader-common-follower games. Optim. Methods Softw. 25(4), 601–623 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Luna, J.P., Sagastizabal, C., Solodov, M.: A class of Dantzig–Wolfe type decomposition methods for variational inequalty problems. Math. Program. 143(1), 177–209 (2014)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mathiesen, L.: An algorithm based on a sequence of linear complementarity problems applied to a Walrasian equilibrium model: an example. Math. Program. 37(1), 1–18 (1987)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Murphy, F.H., Sherali, H.D., Soyster, A.L.: A mathematical programming approach for determining oligopolistic market equilibrium. Math. Program. 24(1), 92–106 (1982)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic Game Theory. Cambridge University Press, New York (2007)zbMATHGoogle Scholar
  25. 25.
    Outrata, J.V., Zowe, J.: A Newton method for a class of quasi-variational inequalities. Comput. Optim. Appl. 4(1), 5–21 (1995)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Philpott, A., Ferris, M., Wets, R.: Equilibrium, uncertainty and risk in hydro-thermal electricity systems. Math. Program. 157(2), 483–513 (2016)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Robinson, S.M.: Equations on monotone graphs. Math. Program. 141(1), 49–101 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Rosen, J.B.: Existence and uniqueness of equilibrium points for concave N-person games. Econometrica 33(3), 520–534 (1965)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Rutherford, T.F.: Extension of GAMS for complementarity problems arising in applied economic analysis. J. Econ. Dyn. Control 19(8), 1299–1324 (1995)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Schiro, D.A., Pang, J.S., Shanbhag, U.V.: On the solution of affine generalized Nash equilibrium problems with shared constraints by Lemke’s method. Math. Program. 142(1), 1–46 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Solodov, M.: Constraint qualifications. In: Cochran, J.J., et al. (ed.), Wiley Encyclopedia of Operations Research and Management Science. Wiley, Inc. (2010)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Mathematics and Computer Science DivisionArgonne National LaboratoryLemontUSA
  2. 2.Department of Computer Sciences and Wisconsin Institute for DiscoveryUniversity of Wisconsin-MadisonMadisonUSA

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