Abstract
We aim at building a bridge between bilevel programming and generalized Nash equilibrium problems. First, we present two Nash games that turn out to be linked to the (approximated) optimistic version of the bilevel problem. Specifically, on the one hand we establish relations between the equilibrium set of a Nash game and global optima of the (approximated) optimistic bilevel problem. On the other hand, correspondences between equilibria of another Nash game and stationary points of the (approximated) optimistic bilevel problem are obtained. Then, building on these ideas, we also propose different Nash-like models that are related to the (approximated) pessimistic version of the bilevel problem. This analysis, being of independent theoretical interest, leads also to algorithmic developments. Finally, we discuss the intrinsic complexity characterizing both the optimistic bilevel and the Nash game models.
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References
F. Caruso, M.B. Lignola, J. Morgan, Regularization and approximation methods in Stackelberg games and bilevel optimization. Technical report, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy, 2019
F.H. Clarke, Y.S. Ledyaev, R.J. Stern, P.R. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178 (Springer, Berlin, 2008)
S. Dempe, B.S. Mordukhovich, A.B. Zemkoho, Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22(4), 1309–1343 (2012)
D. Dorsch, H. Th. Jongen, V. Shikhman, On intrinsic complexity of Nash equilibrium problems and bilevel optimization. J. Optim. Theory Appl. 159, 606–634 (2013)
A. Dreves, C. Kanzow, O. Stein, Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems. J. Global Optim. 53, 587–614 (2012)
F. Facchinei, C. Kanzow, Generalized Nash equilibrium problems. Ann. Oper. Res. 175(1), 177–211 (2010)
H. Th. Jongen, V. Shikhman, Bilevel optimization: on the structure of the feasible set. Math. Program. 136, 65–89 (2012)
H. Th. Jongen, P. Jonker, F. Twilt, Critical sets in parametric optimization. Math. Program. 34, 333–353 (1986)
L. Lampariello, S. Sagratella, A bridge between bilevel programs and Nash games. J. Optim. Theory Appl. 174(2), 613–635 (2017)
L. Lampariello, S. Sagratella, Numerically tractable optimistic bilevel problems. Comput. Optim. Appl. 76, 277–303 (2020)
L. Lampariello, S. Sagratella, O. Stein, The standard pessimistic bilevel problem. SIAM J. Optim. 29, 1634–1656 (2019)
M.B. Lignola, J. Morgan, Topological existence and stability for Stackelberg problems. J. Optim. Theory Appl. 84(1), 145–169 (1995)
M.B. Lignola, J. Morgan, Stability of regularized bilevel programming problems. J. Optim. Theory Appl. 93(3), 575–596 (1997)
G-H. Lin, M. Xu, J.J. Ye, On solving simple bilevel programs with a nonconvex lower level program. Math. Program. 144(1–2), 277–305 (2014)
P. Loridan, J. Morgan, A theoretical approximation scheme for Stackelberg problems. J. Optim. Theory Appl. 61(1), 95–110 (1989)
L. Mallozzi, R. Messalli, S. Patrì, A. Sacco, Some Aspects of the Stackelberg Leader/Follower Model (Springer, Berlin, 2018), pp. 171–181
D.A. Molodtsov, V.V., Fedorov, Approximation of two-person games with information exchange. USSR Comput. Math. Math. Phys. 13(6), 123–142 (1973)
J.V. Outrata, On the numerical solution of a class of Stackelberg problems. Z. Oper. Res. 34(4), 255–277 (1990)
R.T. Rockafellar, J.B. Wets, Variational Analysis (Springer, Berlin, 1998)
O. Stein, Bi-Level Strategies in Semi-Infinite Programming (Kluwer Academic Publishers, Boston, 2003)
O. Stein, How to solve a semi-infinite optimization problem. Eur. J. Oper. Res. 23, 312–320 (2012)
T. Tanino, T. Ogawa, An algorithm for solving two-level convex optimization problems. Int. J. Syst. Sci. 15(2), 163–174 (1984)
A. von Heusinger, C. Kanzow, Optimization reformulations of the generalized Nash equilibrium problem using Nikaido–Isoda-type functions. Comput. Optim. Appl. 43, 353–377 (2009)
J.J. Ye, D.L. Zhu, Optimality conditions for bilevel programming problems. Optimization 33(1), 9–27 (1995)
J.J. Ye, D.L. Zhu, New necessary optimality conditions for bilevel programs by combining the MPEC and value function approaches. SIAM J. Optim. 20(4), 1885–1905 (2010)
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Lampariello, L., Sagratella, S., Shikhman, V., Stein, O. (2020). Interactions Between Bilevel Optimization and Nash Games. In: Dempe, S., Zemkoho, A. (eds) Bilevel Optimization. Springer Optimization and Its Applications, vol 161. Springer, Cham. https://doi.org/10.1007/978-3-030-52119-6_1
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DOI: https://doi.org/10.1007/978-3-030-52119-6_1
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