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The Gauss–Seidel method for generalized Nash equilibrium problems of polynomials

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Abstract

This paper concerns the generalized Nash equilibrium problem of polynomials (GNEPP). We apply the Gauss–Seidel method and Moment-SOS relaxations to solve GNEPPs. The convergence of the Gauss–Seidel method is known for some special GNEPPs, such as generalized potential games (GPGs). We give a sufficient condition for GPGs and propose a numerical certificate, based on Putinar’s Positivstellensatz. Numerical examples for both convex and nonconvex GNEPPs are given for demonstrating the efficiency of the proposed method.

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References

  1. Anselmi, J., Ardagna, D., Passacantando, M.: Generalized nash equilibria for saas/paas clouds. Eur. J. Oper. Res. 236(1), 326–339 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Ardagna, D., Ciavotta, M., Passacantando, M.: Generalized Nash equilibria for the service provisioning problem in multi-cloud systems. IEEE Trans. Serv. Comput. 10, 381–395 (2017)

    Google Scholar 

  3. Arrow, K., Debreu, G.: Existence of an equilibrium for a competitive economy. Econ. J. Econom. Soc. 22, 265–290 (1954)

    MathSciNet  MATH  Google Scholar 

  4. Aubin, J., Frankowska, H.: Set-Valued Analysis. Springer, Berlin (2009)

    MATH  Google Scholar 

  5. Aussel, D., Sagratella, S.: Sufficient conditions to compute any solution of a quasivariational inequality via a variational inequality. Math. Methods Oper. Res. 85(1), 3–18 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-Linear Parametric Optimization. Akademie-Verlag, Berlin (1982)

    MATH  Google Scholar 

  7. Basar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. SIAM 23, (1999)

  8. Breton, M., Zaccour, G., Zahaf, M.: A game-theoretic formulation of joint implementation of environmental projects. Eur. J. Oper. Res. 168, 221–239 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Contreras, J., Klusch, M., Krawczyk, J.B.: Numerical solutions to Nash–Cournot equilibria in coupled constraint electricity markets. IEEE Trans. Power Syst. 19(1), 195–206 (2004)

    Google Scholar 

  10. Couzoudis, E., Renner, P.: Computing generalized Nash equilibria by polynomial programming. Math. Methods Oper. Res. 77(3), 459–472 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Debreu, G.: A social equilibrium existence theorem. Proc. Natl. Acad. Sci. 38, 886–893 (1952)

    MathSciNet  MATH  Google Scholar 

  12. Dontchev, A., Rockafellar, R.T.: Implicit Functions and Solution Mappings, Springer Monogr. Math., (2009)

  13. 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, 63–84 (2014)

    MathSciNet  MATH  Google Scholar 

  14. 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)

    MathSciNet  MATH  Google Scholar 

  15. Facchinei, F., Fischer, A., Piccialli, V.: On generalized Nash games and variational inequalities. Oper. Res. Lett. 35, 159–164 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Facchinei, F., Fischer, A., Piccialli, V.: Generalized Nash equilibrium problems and Newton methods. Math. Program. 117, 163–194 (2009)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. Facchinei, F., Lampariello, L.: Partial penalization for the solution of generalized Nash equilibrium problems. J. Glob. Optim. 50(1), 39–57 (2011)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  21. Fialkow, L., Nie, J.: The truncated moment problem via homogenization and flat extensions. J. Funct. Anal. 263(6), 1682–1700 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  23. Fukushima, M.: Restricted generalized Nash equilibria and controlled penalty algorithm. Comput. Manag. Sci. 8, 201–208 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Harker, P.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)

    MATH  Google Scholar 

  25. Helton, J.W., Nie, J.: A semidefinite approach for truncated K-moment problems. Found. Comput. Math. 12(6), 851–881 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Henrion, D., Lasserre, J.: Detecting global optimality and extracting solutions in GloptiPoly. Positive polynomials in control, 293-C310, Lecture Notes in Control and Inform. Sci., 312, Springer, Berlin (2005)

  27. Henrion, D., Lasserre, J., Löfberg, J.: Gloptipoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Ichiishi, T.: Game Theory for Economic Analysis. Elsevier, Amsterdam (2014)

    MATH  Google Scholar 

  29. Kanzow, C., Steck, D.: Augmented Lagrangian methods for the solution of generalized Nash equilibrium problems. SIAM J. Optim. 26, 2034–2058 (2016)

    MathSciNet  MATH  Google Scholar 

  30. Kesselman, A., Leonardi, S., Bonifaci, V.: Game-theoretic analysis of internet switching with selfish users. Int. Workshop Internet Netw. Econ. 26, 236–245 (2005)

    Google Scholar 

  31. Lasserre, J.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)

    MathSciNet  MATH  Google Scholar 

  32. Lasserre, J.: An Introduction to Polynomial and Semi-Algebraic Optimization, vol. 52. Cambridge University Press, Cambridge (2015)

    MATH  Google Scholar 

  33. Lofberg, J.: YALMIP: A toolbox for modeling and optimization in MATLAB. In: 2004 IEEE international conference on robotics and automation (IEEE Cat. No. 04CH37508), IEEE (2004)

  34. McKenzie, L.: On the existence of a general equilibrium for a competitive market. Econometrica 27, 54–C71 (1959)

    MathSciNet  MATH  Google Scholar 

  35. Monderer, D., Shapley, L.: Potential games. Games Econ. Behav. 14, 124–143 (1996)

    MathSciNet  MATH  Google Scholar 

  36. Morgan, J., Scalzo, V.: Pseudocontinuous functions and existence of Nash equilibria. J. Math. Econ. 43, 174–183 (2007)

    MathSciNet  MATH  Google Scholar 

  37. 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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  39. Nash, J.: Non-cooperative games. Ann. Math. 54, 286–295 (1951)

    MathSciNet  MATH  Google Scholar 

  40. Neel, J., Reed, J., Gilles, R.: The role of game theory in the analysis of software radio networks. In: Proceedings of SDR forum technical conference, 2250–2255 (2002)

  41. Nie, J.: Tight relaxations for polynomial optimization and lagrange multiplier expressions. Math. Program. 178(1–2), 1–37 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Nie, J.: Certifying convergence of Lasserre’s hierarchy via flat truncation. Math. Program. 142(1–2), 485–510 (2013)

    MathSciNet  MATH  Google Scholar 

  43. Nie, J.: Optimality conditions and finite convergence of Lasserre’s hierarchy. Math. Program. 146(1–2), 97–121 (2014)

    MathSciNet  MATH  Google Scholar 

  44. Nie, J.: Linear optimization with cones of moments and nonnegative polynomials. Math. Program. 153(1), 247–274 (2015)

    MathSciNet  MATH  Google Scholar 

  45. Oggioni, G., Smeers, Y., Allevi, E., Schaible, S.: A generalized nash equilibrium model of market coupling in the european power system. Netw. Spatial Econ. 12, 503–560 (2012)

    MathSciNet  MATH  Google Scholar 

  46. Pang, J., Fukushima, M.: Quasi-variational inequalities, generalized nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005)

    MathSciNet  MATH  Google Scholar 

  47. Paccagnan, D., Gentile, B., Parise, F., Kamgarpour, M., Lygeros, J.: Distributed computation of generalized Nash equilibria in quadratic aggregative games with affine coupling constraints. In: IEEE 55th Conference on Decision and Control (CDC), pp. 6123–6128, IEEE, (2016)

  48. Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993)

    MathSciNet  MATH  Google Scholar 

  49. Robinson, S.: Shadow prices for measures of effectiveness, I: Linear model. Oper. Res. 41, 518–535 (1993)

    MathSciNet  MATH  Google Scholar 

  50. Robinson, S.: Shadow prices for measures of effectiveness, II: General model. Oper. Res. 41, 536–548 (1993)

    MathSciNet  MATH  Google Scholar 

  51. Rockafellar, R., Wets, R.: Variational Analysis, vol. 317. Springer, Berlin (2009)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  53. Sagratella, S.: Computing all solutions of Nash equilibrium problems with discrete strategy sets. SIAM J. Optim. 26(4), 2190–2218 (2016)

    MathSciNet  MATH  Google Scholar 

  54. Sagratella, S.: Computing equilibria of Cournot oligopoly models with mixed-integer quantities. Math. Methods Oper. Res. 86(3), 549–565 (2017)

    MathSciNet  MATH  Google Scholar 

  55. Sagratella, S.: Algorithms for generalized potential games with mixed-integer variables. Comput. Optim. Appl. 68(3), 689–717 (2017)

    MathSciNet  MATH  Google Scholar 

  56. Sagratella, S.: On generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables. Optimization 68(1), 197–226 (2019)

    MathSciNet  MATH  Google Scholar 

  57. Scotti, S.: Structural design using equilibrium programming formulations, Ph.D Thesis, (1995)

  58. Sturm, J.: Using sedumi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999)

    MathSciNet  MATH  Google Scholar 

  59. Sun, L., Gao, Z.: An equilibrium model for urban transit assignment based on game theory. Eur. J. Oper. Res. 181, 305–314 (2007)

    MATH  Google Scholar 

  60. von Heusinger, A., Kanzow, C.: Optimization reformulations of the generalized Nash equilibrium problem using Nikaido–Isoda-type functions. Comput. Optim. Appl. 43, 353–377 (2009)

    MathSciNet  MATH  Google Scholar 

  61. von Heusinger, A., Kanzow, C.: Relaxation methods for generalized nash equilibrium problems with inexact line search. J. Optim. Theory Appl. 143, 159–183 (2009)

    MathSciNet  MATH  Google Scholar 

  62. Yin, H., Shanbhag, U.V., Mehta, P.G.: Nash equilibrium problems with congestion costs and shared constraints. In: Proceedings of the 48h IEEE Conference on Decision and Control (CDC), held jointly with 2009 28th Chinese Control Conference (pp. 4649-4654). IEEE

  63. Yue, D., You, F.: Game-theoretic modeling and optimization of multi-echelon supply chain design and operation under stackelberg game and market equilibrium. Comput. Chem. Eng. 71, 347–361 (2014)

    Google Scholar 

  64. Zhou, J., Lam, W., Heydecker, B.: The generalized Nash equilibrium model for oligopolistic transit market with elastic demand. Transp. Res. Part B Methodol. 39, 519–544 (2005)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA

    Jiawang Nie & Xindong Tang

  2. School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing, Jiangsu, 210023, China

    Lingling Xu

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  1. Jiawang Nie
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Nie, J., Tang, X. & Xu, L. The Gauss–Seidel method for generalized Nash equilibrium problems of polynomials. Comput Optim Appl 78, 529–557 (2021). https://doi.org/10.1007/s10589-020-00242-7

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