Summer School

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).

Monday, June 15, 2026

Tuesday, June 16, 2026

* During the informal “free discussion” session, participants will be able to use the lecture hall to review the lectures, discuss, and interact with each other and with the speakers.

Speakers and abstracts

• Margarida Carvalho, Université de Montréal 
  Lecture 1: Duality-based reformulations in bilevel optimization - Part I
  Lecture 2: Duality-based reformulations in bilevel optimization - Part II

  Abstract: 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 complexity

  Abstract: 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

  Abstract: We first discuss the qubit assignment and routing problem, a crucial step in quantum compiling that arises when a logical quantum circuit must be mapped onto hardware with restricted two-qubit connectivity. This problem can be viewed as a combinatorial reconfiguration problem on a graph, where qubits must be moved around to enable the execution of a quantum circuit. We present integer programming models for this problem and report numerical experiments on several hardware topologies that outperform SABRE heuristic, the standard heuristic in Qiskit.
  Motivated by this setting, we then study a related and "simpler" reconfiguration problem known as parallel token swapping, which seeks an optimal sequence of vertex-disjoint swaps to transform an initial token configuration on a graph into a desired one. We also analyze the stretch factor of a natural lower bound for this problem, which has been shown to be useful in the design of heuristics for qubit routing.

  Lecture 2: Mixing set and its extensions

  Abstract: Mixing sets were originally introduced in the context of lot-sizing problems and later extended to the general integer setting. Since then, they have been generalized along multiple dimensions, giving rise to a rich class of mixed-integer sets with well-structured polyhedral descriptions. In the classical setting, mixing inequalities provide a complete linear description of the convex hull and are facet-defining under appropriate conditions. Moreover, they are closely connected to mixed-integer rounding (MIR) and split cuts, and admit efficient separation procedures. These sets arise naturally in applications, particularly in robust and stochastic integer programming. We review the polyhedral characterization of mixing sets and their extensions, including connections to polymatroid inequalities, and highlight recent developments, including a recent application to the boxing problem.

• Hexaly (our Gold Sponsor)
  Title: Hexaly, a global hybrid optimization solver

  Abstract: Hexaly is a new type of global optimization solver. Its modeling formalism is hybrid, unifying concepts from mixed-integer programming, nonlinear programming, constraint programming, and black-box optimization. As a result, it is nonlinear, set-oriented, and supports user-coded functions, enabling seamless integration of simulation with optimization or machine learning with optimization. A central design objective is modeling expressiveness and openness: enabling users to formulate problems closer to their natural combinatorial structure, while allowing diverse algorithmic components to interact in a complementary manner. In this talk, we will present the guiding principles behind this approach, with a particular focus on discrete optimization problems where Hexaly currently demonstrates its strongest performance, such as large‑scale routing, scheduling, and packing. We will also explain how Hexaly combines exact and heuristic optimization methods: spatial branch-and-bound, simplex methods, interior-point methods, constraint propagation, automatic branch-cut-price, local search, surrogate modeling, among others.