NCSU Differential Equations/Nonlinear Analysis Seminars

Location and Time: SAS 4201, Wednesday 15:00 -16:00

Organizers: Lorena Bociu, Patrick Combettes, Ryan Murray, and Khai T. Nguyen

NEXT TALK

Wednesday, Jan 22, 15:00-16:00, Zoom meeting: Link

Speaker: David Salas, Universidad de O’Higgins, Chile
Title: The Bayesian approach for bilevel programming
Abstract: In this talk, we present the Bayesian approach for bilevel programming problems, first introduced by Mallozzi and Morgan [1] under the name of intermediate Stackelberg games, and rediscovered in [3]. The main idea of this approach is to treat the (possibly not single-valued) response set of the follower as an uncertainty set, and treat the actual response as a random variable following a decision-dependent distribution. This distribution models the belief of the leader, hence the name of the approach. In the first part of the talk we will explore the well-posedness of the model, focusing on sufficient conditions to guarantee the existence of optimal solutions. In particular, we will show that in the linear setting, the existence of solutions is guaranteed for a large family of beliefs derived from the natural adaptation of the uniform distribution. In the second part, we will study an application to stochastic linear bilevel optimization, where we leverage on the polyhedral structure of the problem which is revealed under the lens of the Bayesian approach. We will finish discussing some open problems and new directions of research in this topic. The talk is based on the recent contributions with A. Svensson and G. Muñoz  [2,3].

References:
[1] L. Mallozzi and J. Morgan. Hierarchical Systems with Weighted Reaction Set, pages 271–282. Springer US, Boston, MA, 1996.
[2] G. Muñoz, D. Salas, and A. Svensson. Exploiting the polyhedral geometry of stochastic linear bilevel programming. Math. Program., pages 1–36, 2024.
[3] D. Salas and A. Svensson. Existence of solutions for deterministic bilevel games under a general Bayesian approach. SIAM J. Optim., 33(3):2311–2340, 2023.

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