1. Bilevel equilibrium problems 

Orestes Bueno | Universidad del Pacífico

Resumen

We study the existence of solutions for  bilevel equilibrium problems. First, we show that many problems such as multi-leader-follower games can be reformulated in this format.

2. Projected solutions for generalized Nash games

Carlos Calderón | Instituto de Matemática y Ciencias Afines

Resumen

In this work, we focus on the concept of projected solutions for generalized Nash games. We explore the relationship between projected solutions and classical solutions, examining their connections and differences.

3. Lower semicontinuity of intersections of set-valued maps and applications on bilevel games

Anton Svensson | Universidad de O'Higgins

Resumen

We discuss about the lower semicontinuity of set-valued maps, which is a crucial concept in parametric optimization and game theory. The focus is on the intersection of set-valued maps and the preservation of lower semicontinuity under this operation. We present new results based on some minimal properties that ensure the lower semicontinuity of the intersection with other lower semicontinuous maps. Additionally, the lower semicontinuity of the intersection of an infinite family of set-valued maps is considered. We provide an example to illustrate the application of this concept in bilevel games.

4. Exploiting the polyhedral geometry of stochastic linear bilevel programming

David Salas | Universidad de O'Higgins

Resumen

We will study linear bilevel programming problems whose lower-level objective is given by a random cost vector with known distribution. We consider the case where this distribution is nonatomic, allowing to reformulate the problem of the leader using the Bayesian approach previously developed in (Salas and Svensson, SIAM J Optim 33(3):2311– 2340, 2023), with a decision-dependent distribution that concentrates on the vertices of the feasible set of the follower’s problem. We call this a vertex-supported belief. We prove that this formulation is piecewise affine over the so-called chamber complex of the feasible set of the high-point relaxation. We propose two algorithmic approaches to solve general problems enjoying this last property. The first one is based on enu- merating the vertices of the chamber complex. This approach is not scalable, but we present it as a computational baseline and for its theoretical interest. The second one is a Monte-Carlo approximation scheme based on the fact that randomly drawn points of the domain lie, with probability 1, in the interior of full-dimensional chambers, where the problem (restricted to this chamber) can be reduced to a linear program. Finally, we will discuss some perspectives and challenges.