We study connections between optimistic bilevel programming problems and generalized Nash equilibrium problems. We remark that, with respect to bilevel problems, we consider the general case in which the lower level program is not assumed to have a unique solution. Inspired by the optimal value approach, we propose a Nash game that, transforming the so-called implicit value function constraint into an explic- itly defined constraint function, incorporates some taste of hierarchy and turns out to be related to the bilevel programming problem. We provide a complete theoretical analysis of the relationship between the vertical bilevel problem and our “uneven” horizontal model: in particular, we define classes of problems for which solutions of the bilevel program can be computed by finding equilibria of our game. Furthermore, by referring to some applications in economics, we show that our “uneven” horizontal model, in some sense, lies between the vertical bilevel model and a “pure” horizontal game.

A Bridge Between Bilevel Programs and Nash Games / Lampariello, Lorenzo; Sagratella, Simone. - In: JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS. - ISSN 0022-3239. - STAMPA. - (2017), pp. 613-635. [10.1007/s10957-017-1109-0]

A Bridge Between Bilevel Programs and Nash Games

LAMPARIELLO, LORENZO;SAGRATELLA, SIMONE
2017

Abstract

We study connections between optimistic bilevel programming problems and generalized Nash equilibrium problems. We remark that, with respect to bilevel problems, we consider the general case in which the lower level program is not assumed to have a unique solution. Inspired by the optimal value approach, we propose a Nash game that, transforming the so-called implicit value function constraint into an explic- itly defined constraint function, incorporates some taste of hierarchy and turns out to be related to the bilevel programming problem. We provide a complete theoretical analysis of the relationship between the vertical bilevel problem and our “uneven” horizontal model: in particular, we define classes of problems for which solutions of the bilevel program can be computed by finding equilibria of our game. Furthermore, by referring to some applications in economics, we show that our “uneven” horizontal model, in some sense, lies between the vertical bilevel model and a “pure” horizontal game.
2017
bilevel programming; hierarchical optimization problem; Stackelberg game; control and optimization; management science and operations research; applied mathematics
01 Pubblicazione su rivista::01a Articolo in rivista
A Bridge Between Bilevel Programs and Nash Games / Lampariello, Lorenzo; Sagratella, Simone. - In: JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS. - ISSN 0022-3239. - STAMPA. - (2017), pp. 613-635. [10.1007/s10957-017-1109-0]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/973723
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