Skip to main content
SpringerLink
Log in

Non-cooperative games with minmax objectives

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We consider noncooperative games where each player minimizes the sum of a smooth function, which depends on the player, and of a possibly nonsmooth function that is the same for all players. For this class of games we consider two approaches: one based on an augmented game that is applicable only to a minmax game and another one derived by a smoothing procedure that is applicable more broadly. In both cases, centralized and, most importantly, distributed algorithms for the computation of Nash equilibria can be derived.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berinde, V.: Iterative Approximation of Fixed Points. Lectures Notes in Mathematics, vol. 1912. Springer, Berlin (2007)

    Google Scholar 

  2. Bertsekas, D.P.,Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific (1997) (Originally published by Prentice-Hall Inc, in 1989)

  3. Burke, J.V., Hoheisel, T., Kanzow, C.: Gradient consistency for integral-convolution smoothing functions. Preprint 309, University of Würzburg, Insitute of Mathematics, Würzburg, Germany (2012)

  4. Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control. Birkhäuser, Boston (2004)

    MATH  Google Scholar 

  5. Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Progr. 134, 71–99 (2012)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Eaves, B.C.: Polymatrix games with joint constraints. SlAM J. Appl. Math. 24, 418–423 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ermoliev, Y.M., Norkin, V.I., Wets, R.J.B.: The minimization of semicontinuous functions: molifier subgradients. SIAM J. Control Optim. 33, 149–167 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. 4OR 5, 173–210 (2007)

    Google Scholar 

  10. Facchinei, F., Kanzow, C.: Penalty methods for the solution of generalized Nash equilibrium problems. SIAM J. Optim. 20, 2228–2253 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  12. Facchinei, F., Pang, J.S.: Nash equilibria: the variational approach. In: Eldar, Y., Palomar, D. (eds.) Convex Optimization in Signal Processing and Communications, pp. 443–493. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  13. Facchinei, F., Pang, J.-S., Scutari, G., Lampariello, L.: VI-constrained hemivariational inequalities: distributed algorithms and power control in ad-hoc networks. Math. Progr. (2013). doi:10.1007/s10107-013-0640-5. In print. Published electronically

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Fukushima, M.: Restricted generalized Nash equilibria and controlled penalty algorithm. Comput. Manag. Sci. 8, 201–218 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gürkan, G., Pang, J.-S.: Approximations of Nash equilibria. Math. Progr. 117, 223–253 (2009)

    Article  MATH  Google Scholar 

  17. Haurie, A., Krawczyk, J.B.: Optimal charges on river effluent from lumped and distributed sources. Environ. Model. Assess. 2, 93–106 (1997)

    Article  Google Scholar 

  18. von Heusinger, A., Kanzow, C., Fukushima, M.: Newton’s Method for computing a normalized equilibrium in the generalized Nash game through fixed point formulation. Math. Progr. 132, 99–123 (2012)

    Article  MATH  Google Scholar 

  19. Hirsch, F.: Differential Topology. Springer, New York (1976)

    Book  MATH  Google Scholar 

  20. Jofré, A., Wets, R.J.-B.: Variational convergence of bivariate functions: iopsided convergence. Math. Progr. 115, 275–295 (2009)

    Article  Google Scholar 

  21. Kannan, A., Shanbhag, U.V., Kim, H.M.: Addressing supply-side risk in uncertain power markets: stochastic generalized Nash models and scalable algorithms. Optim. Methods Softw. 28, 1095–1138 (2013)

    Google Scholar 

  22. Kannan, A., Shanbhag, U.V., Kim, H.M.: Strategic behavior in power markets under uncertainty. Energy Syst. 2, 115–141 (2011)

    Article  Google Scholar 

  23. Krawczyk, J.B.: Coupled constraint Nash equilibria in environmental games. Resour. Energy Econ. 27, 157–181 (2005)

    Article  Google Scholar 

  24. Krawcyzk, J.B.: Numerical solutions to couple-constraint (or generalized) Nash equilibrium problems. Comput. Manag. Sci. 4, 183–204 (2007)

    Article  MathSciNet  Google Scholar 

  25. Krawczyk, J.B., Uryasev, S.: Relaxation algorithms to find Nash equilibria with economic applications. Environ. Model. Assess. 5, 63–73 (2000)

    Article  Google Scholar 

  26. Kubota, K., Fukushima, M.: Gap function approach to the generalized Nash equilibrium problem. J. Optim. Theory Appl. 144, 511–531 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kulkarni, A.A., Shanbhag, U.V.: On the variational equilibrium as a refinement of the generalized Nash equilibrium. Automatica 48, 45–55 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kulkarni, A.A., Shanbhag, U.V.: Revisiting generalized Nash games and variational inequalities. J. Optim. Theory Appl. 154, 175–186 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lakshmanan, H., Farias, D.: Decentralized resource allocation in dynamic networks of agents. SIAM J. Optim. 19, 911–940 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Nabetani, K., Tseng, P., Fukushima, M.: Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints. Comput. Optim. Appl. 48, 423–452 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  31. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005). Erratum, ibid. 6 (2009) 373–375

    Article  MATH  MathSciNet  Google Scholar 

  32. Pang, J.S., Scutari, G.: Nonconvex games with side constraints. SIAM J. Optim. 21, 1491–1522 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  33. Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)

    Google Scholar 

  34. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  35. Rosen, J.: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33, 520–534 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  36. Scutari, G., Facchinei, F., Pang, J.-S., Palomar, D.: Real and complex monotone communication games. IEEE Trans Inform Theory (submitted). Technical report (2013). Available at http://arxiv.org/abs/1212.6235

  37. Scutari, G., Facchinei, F., Song, P., Palomar, D., Pang, J.-S.: Decomposition by partial linearization: parallel optimization of multi-agent systems. IEEE Trans. Signal Process. 62, 641–656 (2014)

    Article  MathSciNet  Google Scholar 

  38. Steklov, V.A.: Sur les expressions asymptotiques de certaines fonctions definies par les équations differentielles du second ordre et leurs applications au problème du dévelopement d’une fonction arbitraire en séries procédant suivant les diverses fonctions. Commun. Kharkov Math. Soc. 2, 97–199 (1907)

    Google Scholar 

  39. Uryasev, S., Rubinstein, R.Y.: On relaxation algorithms in computation of noncooperative equilibria. IEEE Trans. Autom. Control 39, 1263–1267 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  40. Vial, J.-P.: Strong and weak convexity of sets and functions. Math. Oper. Res. 8, 231–259 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  41. Yousefian, F., Nedich, A., Shanbhag, U.V.: On stochastic gradient and subgradient methods with adaptive steplength sequences. Automatica 48, 56–67 (2012)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors thank two referees who have offered constructive comments that have improved the presentation of the paper. In particular, one referee has provided additional references, including [38], that are related to our work. The work of the second author was based on research supported by the U.S.A. National Science Foundation Grant CMMI–0969600 while the work of the third author was based on research supported by the U.S.A. National Science Foundation CNS–1218717 and CAREER Grant #1254739

Author information

Authors and Affiliations

  1. Department of Computer, Control, and Management Engineering Antonio Ruberti, University of Rome La Sapienza, Via Ariosto 25, 00185 , Rome, Italy

    Francisco Facchinei

  2. Deptartment of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA, USA

    Jong-Shi Pang

  3. Department of Electrical Engineering, State University of New York (SUNY) at Buffalo, Buffalo, NY, USA

    Gesualdo Scutari

Authors
  1. Francisco Facchinei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jong-Shi Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gesualdo Scutari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco Facchinei.

Additional information

To Masao Fukushima, on the occasion of his 65th birthday, with friendship and admiration.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Facchinei, F., Pang, JS. & Scutari, G. Non-cooperative games with minmax objectives. Comput Optim Appl 59, 85–112 (2014). https://doi.org/10.1007/s10589-014-9642-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-014-9642-3

Keywords

Search

Navigation