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Computing generalized Nash equilibria by polynomial programming

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Abstract

We present a new way to solve generalized Nash equilibrium problems. We assume the feasible set to be compact. Furthermore all functions are assumed to be polynomials. However we do not impose convexity on either the utility functions or the action sets. The key idea is to use Putinar’s Positivstellensatz, a representation result for positive polynomials, to replace each agent’s problem by a convex optimization problem. The Nash equilibria are then feasible solutions to a system of polynomial equations and inequalities. Our application is a model of the New Zealand electricity spot market with transmission losses based on a real dataset.

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Notes

  1. Electricity Authority NZ (2012).

  2. Electricity Authority NZ (2012).

  3. (Electricity Authority NZ 2010b, 13.6).

  4. (Electricity Authority NZ 2010b, 13.9, 13.12).

  5. (Electricity Authority NZ 2010b, 13.16).

  6. (Electricity Authority NZ 2010b, 13.15).

  7. See e.g. Boyd and Vandenberghe (2004).

References

  • Aussel D, Correa R, Marechal M (2011a) Gap functions for quasivariational inequalities and generalized nash equilibrium problems. J Optim Theory Appl 151(3):474–488

    Article  MathSciNet  MATH  Google Scholar 

  • Aussel D, Correa R, Marechal P (2011b) Spot electricity market with transmission losses

  • Aussel D, Dutta J (2011) On gap functions for multivalued stampacchia variational inequalities. J Optim Theory Appl 149(3):513–527

    Article  MathSciNet  MATH  Google Scholar 

  • Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Electricity Authority NZ (2010a) GEM: generation expansion model. Accessed: 12.10.2011

  • Electricity Authority NZ (2010b) Electricity industry participation code 2010 - Part 13. Version: 21.09.2012

  • Electricity Authority NZ (2011) CDS: centralised dataset. Accessed: 01.10.2012

  • Electricity Authority NZ (2012) Market operation service providers. http://www.ea.govt.nz/industry/mo-service-providers/, Accessed: 01.10.2012

  • Facchinei F, Kanzow C (2007) Generalized nash equilibrium problems. 4OR Q J Oper Res 5(3):173–210

    Article  MathSciNet  MATH  Google Scholar 

  • Henrion D, Lasserre J, Löfberg J (2007) GloptiPoly 3: moments, optimization and semidefinite programming. Optim Methods Softw 24(4–5):761–779

    Google Scholar 

  • Hobbs B, Metzler C, Pang J-S (2000) Strategic gaming analysis for electric power systems: an mpec approach. IEEE Trans Power Syst 15(2):638–645

    Article  Google Scholar 

  • Jibetean D, de Klerk E (2006) Global optimization of rational functions: a semidefinite programming approach. Math. Program. 106:93–109

    Article  MathSciNet  MATH  Google Scholar 

  • Lasserre JB (2010a) A “Joint+Marginal” approach to parametric polynomial optimization. SIAM J. Optim. 20:1995–2022

    Article  MathSciNet  MATH  Google Scholar 

  • Lasserre JB (2010b) Moments, positive polynomials and their applications, vol 1 of Imperial College Press Optimization Series. Imperial College Press, London

    Google Scholar 

  • Laurent M (2009) Sums of squares, moment matrices and optimization over polynomials, vol 149 of IMA volumes in Mathematics and its Applications. Springer, Berlin

    Google Scholar 

  • Rosen JB (1965) Existence and uniqueness of equilibrium points for concave N-Person games. Econometrica 33(3):520–534

    Article  MathSciNet  MATH  Google Scholar 

  • Tipler P, Gerlich D, Jerke G (2000) Physik, Volume 3. korr. Nachdr. of Spektrum Lehrbuch. Spektrum Akad. Verl

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Authors and Affiliations

  1. Department of Business Administration, University of Zuerich, Moussonstrasse 15, 8044, Zuerich, Switzerland

    Eleftherios Couzoudis & Philipp Renner

Authors
  1. Eleftherios Couzoudis
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  2. Philipp Renner
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Corresponding author

Correspondence to Eleftherios Couzoudis.

Additional information

We thank Didier Aussel, Hans-Jakob Lüthi, Cordian Riener and two anonymous referees.

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Couzoudis, E., Renner, P. Computing generalized Nash equilibria by polynomial programming. Math Meth Oper Res 77, 459–472 (2013). https://doi.org/10.1007/s00186-012-0422-5

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  • DOI: https://doi.org/10.1007/s00186-012-0422-5

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