Softening bilevel problems via two-scale Gibbs measures

Guillaume Carlier (1) , Lina Mallozzi (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”

We introduce a new, and elementary, approximation method for bilevel optimization problems motivated by Stackelberg leader-follower games. Our technique is based on the notion of two-scale Gibbs measures. The first scale corresponds to the cost function of the follower and the second scale to that of the leader. We explain how to choose the weights corresponding to these two scales under very general assumptions and establish rigorous Γ-convergence results. An advantage of our method is that it is applicable both to optimistic and to pessimistic bilevel problems.

Mots clés: bilevel optimization; Stackelberg games; Gibbs measures; Γ-convergence; Laplace method MS Classification: 49K35; 90C26; 60B05

Domaines: Optimisation et contrôle [math.OC]

Guillaume Carlier, Lina Mallozzi. Softening bilevel problems via two-scale Gibbs measures. Set-Valued and Variational Analysis, 2022, 30 (2), pp.573-595. ⟨10.1007/s11228-021-00605-0⟩. ⟨hal-02305909⟩