The IPCO 2026 Summer School will be held on June 15-16 in room 1A150 of the Department of Mathematics "Tullio Levi-Civita", 63 Via Trieste, Padova. Please note that this venue is distinct from the conference site, which is approximately a 15-minute walk away (see the Local Information page).
The summer school will feature the following three speakers, each delivering two 90-minute lectures:
- Margarida Carvalho, Université de Montréal
Lecture 1: Duality-based reformulations in bilevel optimization - Part I
Lecture 2: Duality-based reformulations in bilevel optimization - Part IIAbstract: Bilevel optimization is a powerful framework for modeling hierarchical decision-making between a leader and one or more followers. These lectures provide a systematic exploration of duality-based reformulations, progressing from classical linear models to complex mixed-integer settings.
In the first lecture, we will begin with an introduction to bilevel optimization, defining key terminology and different solution concepts, and discussing the inherent computational complexity of these problems. We then focus on the linear-follower case, where necessary and sufficient optimality conditions allow for various direct single-level reformulations via KKT conditions or strong duality, utilizing different primal and dual representations of the lower-level problem. From this, we will examine the polyhedral description of the bilevel feasible set and demonstrate how this can be used to tackle a class of integer programming games.
While these reformulations arise naturally in the linear case, they become significantly more challenging when the lower-level problem includes nonlinearities or nonconvexities, such as integrality constraints. We will explore duality-based reformulations in the context of followers modeled via monotropic programming, which arise in user equilibrium problems. Finally, we will review recent research that leverages dynamic programming to devise reformulations in a dualize-and-combine fashion for mixed-integer bilevel programs. This area is still in its early stages, presenting many opportunities for further exploration. - Alberto Del Pia, University of Wisconsin-Madison
Lecture 1: Mixed integer quadratic programming I: Structural properties
Lecture 2: Mixed integer quadratic programming II: Algortihms and complexityAbstract: Mixed Integer Quadratic Programming (MIQP) concerns the minimization of a quadratic function over a polyhedral set with a subset of variables constrained to be integer. It naturally extends Mixed Integer Linear Programming as well as Quadratic Programming and arises in a wide range of applications. In these lectures, we survey selected recent theoretical developments in MIQP, with an emphasis on structural properties, computational complexity, and algorithmic approaches.
The first lecture focuses on fundamental structural aspects of MIQP, including attainability of optimal solutions, rationality and encoding size, the role of unbounded directions, and connections with complexity classes such as NP. The second lecture turns to algorithmic developments, highlighting recent results on exact and approximation algorithms with provable guarantees. - Oktay Günlük, Georgia Institute of Technology
Lecture 1: Two graph problems related to quantum compiling: Qubit routing and parallel token swapping
Lecture 2: Mixing set and its extensionsAbstract: Coming soon.