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Copositive Duality for Discrete Markets and Games

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  • Cheng Guo
  • Merve Bodur
  • Joshua A. Taylor

Abstract

Optimization problems with discrete decisions are nonconvex and thus lack strong duality, which limits the usefulness of tools such as shadow prices and the KKT conditions. It was shown in Burer(2009) that mixed-binary quadratic programs can be written as completely positive programs, which are convex. Completely positive reformulations of discrete optimization problems therefore have strong duality if a constraint qualification is satisfied. We apply this perspective in two ways. First, we write unit commitment in power systems as a completely positive program, and use the dual copositive program to design a new pricing mechanism. Second, we reformulate integer programming games in terms of completely positive programming, and use the KKT conditions to solve for pure strategy Nash equilibria. To facilitate implementation, we also design a cutting plane algorithm for solving copositive programs exactly.

Suggested Citation

  • Cheng Guo & Merve Bodur & Joshua A. Taylor, 2021. "Copositive Duality for Discrete Markets and Games," Papers 2101.05379, arXiv.org, revised Jan 2021.
    • Handle: RePEc:arx:papers:2101.05379
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    References listed on IDEAS

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    1. Steven Gabriel & Sauleh Siddiqui & Antonio Conejo & Carlos Ruiz, 2013. "Solving Discretely-Constrained Nash–Cournot Games with an Application to Power Markets," Networks and Spatial Economics, Springer, vol. 13(3), pages 307-326, September.
    2. O'Neill, Richard P. & Sotkiewicz, Paul M. & Hobbs, Benjamin F. & Rothkopf, Michael H. & Stewart, William R., 2005. "Efficient market-clearing prices in markets with nonconvexities," European Journal of Operational Research, Elsevier, vol. 164(1), pages 269-285, July.
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    4. Danilov, Vladimir & Koshevoy, Gleb & Murota, Kazuo, 2001. "Discrete convexity and equilibria in economies with indivisible goods and money," Mathematical Social Sciences, Elsevier, vol. 41(3), pages 251-273, May.
    5. Elizabeth Baldwin & Paul Klemperer, 2019. "Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities," Econometrica, Econometric Society, vol. 87(3), pages 867-932, May.
    6. Mallick, Indrajit, 2011. "On the existence of pure strategy Nash equilibria in two person discrete games," Economics Letters, Elsevier, vol. 111(2), pages 144-146, May.
    7. Navid Azizan & Yu Su & Krishnamurthy Dvijotham & Adam Wierman, 2020. "Optimal Pricing in Markets with Nonconvex Costs," Operations Research, INFORMS, vol. 68(2), pages 480-496, March.
    8. David Avis & Gabriel Rosenberg & Rahul Savani & Bernhard Stengel, 2010. "Enumeration of Nash equilibria for two-player games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 9-37, January.
    9. Guanglin Xu & Samuel Burer, 2018. "A copositive approach for two-stage adjustable robust optimization with uncertain right-hand sides," Computational Optimization and Applications, Springer, vol. 70(1), pages 33-59, May.
    10. George Liberopoulos & Panagiotis Andrianesis, 2016. "Critical Review of Pricing Schemes in Markets with Non-Convex Costs," Operations Research, INFORMS, vol. 64(1), pages 17-31, February.
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    Cited by:

    1. Dimitri J. Papageorgiou & Francisco Trespalacios & Stuart Harwood, 2021. "A Note on Solving Discretely-Constrained Nash-Cournot Games via Complementarity," Networks and Spatial Economics, Springer, vol. 21(2), pages 325-330, June.

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