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Revisiting Path-Following to Solve the Generalized Nash Equilibrium Problem

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Recent Developments in Mathematical, Statistical and Computational Sciences (AMMCS 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 343))

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Abstract

In this short paper, we present a generic path-following approach to tackle the generalized Nash equilibrium problem (GNEP) via its KKT conditions. This general formulation can be specialized to various smoothing techniques, including the popular interior-point method. We prove that under classical assumptions, there exists a path starting from an initial point and leading to an equilibrium of the GNEP. We also open the discussion on how one can derive numerical methods based on our approach.

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References

  1. Abdallah, L., Haddou, M., Migot, T.: A sub-additive DC approach to the complementarity problem. Comput. Optim. Appl. 73(2), 509–534 (2019)

    Google Scholar 

  2. Allgower, E.L., Georg, K.: Numerical Continuation Methods: An Introduction, vol. 13. Springer Science & Business Media (2012)

    Google Scholar 

  3. Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econ. J. Econ. Soc. 265–290 (1954)

    Google Scholar 

  4. Aussel, D., Svensson, A.: Towards tractable constraint qualifications for parametric optimisation problems and applications to generalised Nash games. J. Optim. Theory Appl. 182(1), 404–416 (2019)

    Google Scholar 

  5. Bueno, L.F., Haeser, G., Rojas, F.N.: Optimality conditions and constraint qualifications for generalized Nash equilibrium problems and their practical implications. SIAM J. Optim. 29(1):31–54 (2019)

    Google Scholar 

  6. Cavazzuti, E., Pappalardo, M., Passacantando, M.: Nash equilibria, variational inequalities, and dynamical systems. J. Optim. Theory Appl. 114(3), 491–506 (2002)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Cojocaru, M.-G., Wild, E., Small, A.: On describing the solution sets of generalized Nash games with shared constraints. Optim. Eng. 19(4), 845–870 (2018)

    Google Scholar 

  9. Dorsch, D., Jongen, H.T., Shikhman, V.: On structure and computation of generalized Nash equilibria. SIAM J. Optim. 23(1), 452–474 (2013)

    Article  MathSciNet  Google Scholar 

  10. Dreves, A.: Improved error bound and a hybrid method for generalized Nash equilibrium problems. Comput. Optim. Appl. 65(2), 431–448 (2016)

    Article  MathSciNet  Google Scholar 

  11. Dreves, A.: How to select a solution in generalized Nash equilibrium problems. J. Optim. Theory Appl. 178(3), 973–997 (2018)

    Article  MathSciNet  Google Scholar 

  12. Dreves, A., Facchinei, F., Fischer, A., Herrich, M.: A new error bound result for generalized Nash equilibrium problems and its algorithmic application. Comput. Optim. Appl. 59(1), 63–84 (2014)

    Google Scholar 

  13. Dreves, A., Facchinei, F., Kanzow, C., Sagratella, S.: On the solution of the KKT conditions of generalized Nash equilibrium problems. SIAM J. Optim. 21(3), 1082–1108 (2011)

    Google Scholar 

  14. Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175(1), 177–211 (2010)

    Google Scholar 

  15. Facchinei , F., Pang, J.-S.: Exact Penalty Functions for Generalized Nash Problems, pp. 115–126. Springer US, Boston, MA (2006)

    Google Scholar 

  16. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Science & Business Media (2007)

    Google Scholar 

  17. Facchinei, F., Pang, J.-S.: Nash equilibria: the variational approach. In: Convex Optimization in Signal Processing and Communications, pp. 443–493. Cambridge University Press (2010)

    Google Scholar 

  18. Facchinei, F., Piccialli, V., Sciandrone, M.: Decomposition algorithms for generalized potential games. Comput. Optim. Appl. 50(2), 237–262 (2011)

    Google Scholar 

  19. Fan, X., Jiang, L., Li, M.: Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. J. Ind. Manag. Optim. 15(4), 1795–1807 (2019)

    Google Scholar 

  20. Fischer, A., Herrich, M.: Newton-type methods for Fritz John systems of generalized Nash equilibrium problems. Pure Appl. Funt. Anal. 3(4), 587–602 (2018)

    Google Scholar 

  21. Fischer, A., Herrich, M., Izmailov, A.F., Solodov, M.V.: A globally convergent LP-Newton method. SIAM J. Optim. 26(4), 2012–2033 (2016)

    Google Scholar 

  22. Fischer, A., Herrich, M., Schönefeld, K.: Generalized Nash equilibrium problems-recent advances and challenges. Pesqui. Oper. 34(3), 521–558 (2014)

    Google Scholar 

  23. Gauvin, J.: A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Math. Program. 12(1), 136–138 (1977)

    Article  MathSciNet  Google Scholar 

  24. Haddou, M., Migot, T., Omer, J.: A generalized direction in interior point method for monotone linear complementarity problems. Optim. Lett. 13(1), 35–53 (2019)

    Google Scholar 

  25. Izmailov, A.F., Solodov, M.V.: On error bounds and Newton-type methods for generalized Nash equilibrium problems. Comput. Optim. Appl. 59(1), 201–218 (2014)

    Article  MathSciNet  Google Scholar 

  26. Kanzow, C., Steck, D.: Quasi-variational inequalities in Banach spaces: theory and augmented Lagrangian methods. SIAM J. Optim. 29(4), 3174–3200 (2019)

    Google Scholar 

  27. Migot, T., Cojocaru, M.-G.: A decomposition method for a class of convex generalized Nash equilibrium problems. Optim. Eng. (2020)

    Google Scholar 

  28. Migot, T., Cojocaru, M.-G.: Nonsmooth dynamics of generalized Nash games. J. Nonlinear Var. Anal. 1(4), 27–44 (2020)

    Google Scholar 

  29. Migot, T., Cojocaru, M.-G.: A parametrized variational inequality approach to track the solution set of a generalized Nash equilibrium problem. Eur. J. Oper. Res. 283(3), 1136–1147 (2020)

    Google Scholar 

  30. Nash, J.F.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. 36(1), 48–49 (1950)

    Article  MathSciNet  Google Scholar 

  31. Nikaidô, H., Isoda, K.: Note on non-cooperative convex games. Pac. J. Math. 5(Suppl. 1), 807–815 (1955)

    Google Scholar 

  32. Pang, J.-S., Scutari, G., Facchinei, F., Wang, C.: Distributed power allocation with rate constraints in gaussian parallel interference channels. IEEE Trans. Inf. Theory 54(8), 3471–3489 (2008)

    Google Scholar 

  33. Stein, O., Sudermann-Merx, N.: The cone condition and nonsmoothness in linear generalized Nash games. J. Optim. Theory Appl. 170(2), 687–709 (2016)

    Google Scholar 

  34. Terlaky, T.: Interior Point Methods of Mathematical Programming, vol. 5. Springer Science & Business Media (2013)

    Google Scholar 

  35. Zangwill, W.I., Garcia, C.B.: Equilibrium programming: the path following approach and dynamics. Math. Progr. 21(1), 262–289 (1981)

    Google Scholar 

  36. Zangwill, W.I., Garcia, C.B.: Pathways to Solutions, Fixed Points and Equilibria. Prentice-Hall,Englewood Cliffs, NJ (1981)

    Google Scholar 

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Acknowledgements

This work was supported by an NSERC Discovery Accelerator Supplement, grant number 401285 of the second author. The authors would like to thank anonymous referees for their helpful remarks and comments.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Guelph, Guelph, N1G 2W1, Canada

    Tangi Migot & Monica-G. Cojocaru

Authors
  1. Tangi Migot
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  2. Monica-G. Cojocaru
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Corresponding author

Correspondence to Tangi Migot .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada

    D. Marc Kilgour

  2. Department of Mathematics and Statistics, University of Guelph, Guelph, ON, Canada

    Herb Kunze

  3. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada

    Roman Makarov

  4. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada

    Roderick Melnik

  5. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada

    Xu Wang

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Migot, T., Cojocaru, MG. (2021). Revisiting Path-Following to Solve the Generalized Nash Equilibrium Problem. In: Kilgour, D.M., Kunze, H., Makarov, R., Melnik, R., Wang, X. (eds) Recent Developments in Mathematical, Statistical and Computational Sciences. AMMCS 2019. Springer Proceedings in Mathematics & Statistics, vol 343. Springer, Cham. https://doi.org/10.1007/978-3-030-63591-6_9

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