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Regularization and Approximation Methods in Stackelberg Games and Bilevel Optimization

In: Bilevel Optimization

Author

Listed:
  • Francesco Caruso

    (University of Naples Federico II)

  • M. Beatrice Lignola

    (University of Naples Federico II)

  • Jacqueline Morgan

    (University of Naples Federico II)

Abstract

In a two-stage Stackelberg game, depending on the leader’s information about the choice of the follower among his optimal responses, one can associate different types of mathematical problems. We present formulations and solution concepts for such problems, together with their possible roles in bilevel optimization, and we illustrate the crucial issues concerning these solution concepts. Then, we discuss which of these issues can be positively or negatively answered and how managing the latter ones by means of two widely used approaches: regularizing the set of optimal responses of the follower, via different types of approximate solutions, or regularizing the follower’s payoff function, via the Tikhonov or the proximal regularizations. The first approach allows to define different kinds of regularized problems whose solutions exist and are stable under perturbations assuming sufficiently general conditions. Moreover, when the original problem has no solutions, we consider suitable regularizations of the second-stage problem, called inner regularizations, which enable to construct a surrogate solution, called viscosity solution, to the original problem. The second approach permits to overcome the non-uniqueness of the follower’s optimal response, by constructing sequences of Stackelberg games with a unique second-stage solution which approximate in some sense the original game, and to select among the solutions by using appropriate constructive methods.

Suggested Citation

  • Francesco Caruso & M. Beatrice Lignola & Jacqueline Morgan, 2020. "Regularization and Approximation Methods in Stackelberg Games and Bilevel Optimization," Springer Optimization and Its Applications, in: Stephan Dempe & Alain Zemkoho (ed.), Bilevel Optimization, chapter 0, pages 77-138, Springer.
  • Handle: RePEc:spr:spochp:978-3-030-52119-6_4
    DOI: 10.1007/978-3-030-52119-6_4
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    Cited by:

    1. Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2021. "A Local Variation Method for Bilevel Nash Equilibrium Problems," CSEF Working Papers 620, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.
    2. Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2022. "Asymptotic Behavior of Subgame Perfect Nash Equilibria in Stackelberg Games," CSEF Working Papers 661, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.

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