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Existence results for generalized Nash equilibrium problems under continuity-like properties of sublevel sets. (English) Zbl 1476.91010

Summary: A generalized Nash equilibrium problem corresponds to a noncooperative interaction between a finite set of players in which the cost function and the feasible set of each player depend on the decisions of the others. The classical existence result for generalized equilibria due to Arrow and Debreu requires continuity of the cost functions. In this work, we provide an existence of solutions transferring this hypothesis to a “continuity-like” condition over the sublevel sets of the aforementioned functions. Comparison with Reny’s approach for discontinuous games is also considered.

MSC:

91A11 Equilibrium refinements
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[1] M. Ait Mansour and D. Aussel, Quasimonotone variational inequalities and quasiconvex programming: Qualitative stability, J. Convex Anal., 15 (2008), pp. 459-472. · Zbl 1145.49008
[2] S. Al-Homidan, N. Hadjisavvas, and L. Shaalan, Transformation of quasiconvex functions to eliminate local minima, J. Optim. Theory Appl., 177 (2018), pp. 93-105, https://doi.org/10.1007/s10957-018-1223-7. · Zbl 1404.90100
[3] K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica, 22 (1954), pp. 265-290, https://doi.org/10.2307/1907353. · Zbl 0055.38007
[4] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems Control Found. Appl. 2, Birkhäuser Boston, Boston, 1990. · Zbl 0713.49021
[5] D. Aussel, New developments in quasiconvex optimization, in Fixed Point Theory, Variational Analysis, and Optimization, CRC Press, Boca Raton, FL, 2014, pp. 171-205, https://doi.org/10.1201/b17068-8.
[6] D. Aussel, G. Bouza, S. Dempe, and S. Lepaul, Genericity analysis of multi-leader-disjoint-followers game: Theory and applications to contract design in electricity market, SIAM J. Optim., 31 (2021), pp. 2055-2079. · Zbl 1476.91029
[7] D. Aussel, K. Cao Van and D. Salas, Quasi-variational inequality problems over product sets with quasi-monotone operators, SIAM J. Optim., 29 (2019), pp. 1558-1577. · Zbl 1422.49017
[8] D. Aussel, K. Cao Van, and D. Salas, Quasi-variational inequality problems over product sets with quasi-monotone operators, SIAM J. Optim., 29 (2019), pp. 1558-1577, https://doi.org/10.1137/18M1191270. · Zbl 1422.49017
[9] D. Aussel and J. Cotrina, Quasimonotone quasivariational inequalities: Existence results and applications, J. Optim. Theory Appl., 158 (2013), pp. 637-652, https://doi.org/10.1007/s10957-013-0270-3. · Zbl 1284.49014
[10] D. Aussel and J. Cotrina, Stability of quasimonotone variational inequality under sign-continuity, J. Optim. Theory Appl., 158 (2013), pp. 653-667, https://doi.org/10.1007/s10957-013-0272-1. · Zbl 1275.49014
[11] D. Aussel and J. Dutta, Generalized Nash equilibrium problem, variational inequality and quasiconvexity, Oper. Res. Lett., 36 (2008), pp. 461-464, https://doi.org/10.1016/j.orl.2008.01.002. · Zbl 1155.91320
[12] D. Aussel and N. Hadjisavvas, Adjusted sublevel sets, normal operator, and quasi-convex programming, SIAM J. Optim., 16 (2005), pp. 358-367, https://doi.org/10.1137/040606958. · Zbl 1098.49015
[13] D. Aussel and A. Sultana, Quasi-variational inequality problems with non-compact valued constraint maps, J. Math. Anal. Appl., 456 (2017), pp. 1482-1494, https://doi.org/10.1016/j.jmaa.2017.06.034. · Zbl 06864253
[14] D. Aussel and A. Svensson, A short state of the art on multi-leader-follower games, in Bilevel Optimization: Advances and Next Challenges, S. Dempe and A. Zemkoho, eds., Springer, Cham, Switzerland, 2020, pp. 53-76, https://doi.org/10.1007/978-3-030-52119-6_3. · Zbl 1477.91008
[15] G. Beer, R. T. Rockafellar, and R. J.-B. Wets, A characterization of epi-convergence in terms of convergence of level sets, Proc. Amer. Math. Soc., 116 (1992), pp. 753-761. · Zbl 0769.49011
[16] P. Bich and R. Laraki, Externalities in economies with endogenous sharing rules, Econ. Theory Bull., 5 (2017), pp. 127-137, https://doi.org/10.1007/s40505-017-0118-3.
[17] G. Debreu, A social equilibrium existence theorem, Proc. Natl. Acad. Sci. USA, 38 (1952), pp. 886-893, https://doi.org/10.1073/pnas.38.10.886. · Zbl 0047.38804
[18] A. Dreves and J. Gwinner, Jointly convex generalized Nash equilibria and elliptic multiobjective optimal control, J. Optim. Theory Appl., 168 (2016), pp. 1065-1086, https://doi.org/10.1007/s10957-015-0788-7. · Zbl 1337.49051
[19] F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, 4OR, 5 (2007), pp. 173-210, https://doi.org/10.1007/s10288-007-0054-4. · Zbl 1211.91162
[20] N. Hadjisavvas, Continuity and maximality properties of pseudomonotone operators, J. Convex Anal., 10 (2003), pp. 459-469. · Zbl 1063.47041
[21] M. Hintermüller and T. Surowiec, A PDE-constrained generalized Nash equilibrium problem with pointwise control and state constraints, Pac. J. Optim., 9 (2013), pp. 251-273. · Zbl 1269.65062
[22] M. Hintermüller, T. M. Surowiec, and A. Kämmler, Generalized Nash equilibrium problems in Banach spaces: Theory, Nikaido-Isoda-based path-following methods, and applications, SIAM J. Optim., 25 (2015), pp. 1826-1856. · Zbl 1323.65075
[23] C. Kanzow, V. Karl, D. Steck, and D. Wachsmuth, The multiplier-penalty method for generalized Nash equilibrium problems in Banach spaces, SIAM J. Optim., 29 (2019), pp. 767-793. · Zbl 1419.91105
[24] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math., 3 (1969), pp. 510-585, https://doi.org/10.1016/0001-8708(69)90009-7. · Zbl 0192.49101
[25] J. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, Comput. Manag. Sci., 2 (2005), pp. 21-56, https://doi.org/10.1007/s10287-004-0010-0. · Zbl 1115.90059
[26] M. A. Ramos, M. Boix, D. Aussel, L. Montastruc, and S. Domenech, Water integration in eco-industrial parks using a multi-leader-follower approach, Comput. Chem. Eng., 87 (2016), pp. 190-207, https://doi.org/10.1016/j.compchemeng.2016.01.005.
[27] P. J. Reny, Nash equilibrium in discontinuous games, Econom. Theory, 61 (2016), pp. 553-569, https://doi.org/10.1007/s00199-015-0934-3. · Zbl 1367.91019
[28] R. T. Rockafellar and R.-B. Wets, Variational Analysis, Grundlehren Math. Wiss. 317, Springer, Berlin, 1998, https://doi.org/10.1007/978-3-642-02431-3. · Zbl 0888.49001
[29] D. Salas, K. C. Van, D. Aussel, and L. Montastruc, Optimal design of exchange networks with blind inputs and its application to eco-industrial parks, Comput. Chem. Eng., 143 (2020), 107053, https://doi.org/10.1016/j.compchemeng.2020.107053.
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