In this work, we study the Traveling Purchaser Problem with Incompatibility Constraints (TPP-IC), an NP-hard combinatorial optimization problem that extends the classical Traveling Purchaser Problem (TPP) by introducing incompatibilities between items. These prohibit incompatible items from being transported on the same route. We model the TPP-IC using structures called compatibility graphs, which ensure, by construction, that any feasible route does not transport incompatible items. To address the problem, we propose several two-index mixed-integer linear programming (MILP) formulations, including both item-based and market-based approaches. We present a theoretical and computational comparison of these models. In addition, we introduce a family of valid inequalities that exploit the incompatibility constraints. These inequalities strengthen the formulations and are incorporated into a branch-and-cut algorithm. Our computational experiments show that the two-index formulations based on the compatibility graphs yield significantly higher linear programming relaxation values compared to the three-index formulations in the literature. They also solve more benchmark instances. Our results also highlight the main factors that contribute to the difficulty of solving TPP-IC instances and reveal the limitations of exact solution methods in instances with a high degree of item incompatibility.

Optimal transport is a very versatile theory that lifts the geometry on a given underlying space to the overlying Wasserstein space of probability measures. Based on this rich interpretation, Jordan, Kinderlehrer and Otto showed 25 years ago that the (linear!) heat equation can be seen as the (highly nonlinear!) Wasserstein gradient flow of the Boltzmann entropy in the space of probability measures, thus providing a strong form of the second principle of thermodynamics. This sparked tremendous interest due to the interconnection between various mathematical branches, ranging from applied PDEs, numerical analysis, probability and interacting particle systems, metric geometry, functional analysis, and more. In this talk I will try to review the main ideas behind this formalism, and show how classical optimal transport can be extended to study a broad class of evolution equations from a variational standpoint (eg. Hele-Shaw dynamics, reaction-diffusion, multiphase flows, evolutionary genetics, etc.).

A crucial task  in control applications is the design of a feedback operator allowing us to compute a control input which is able to stabilize a given dynamical system, being able to respond to small perturbations as well. Feedback inputs are given as a function of the state of the system, which is often not fully available in real world applications. Thus, another crucial task is the design of a dynamic Luenberger observer providing us with an estimate for the unknown state, by using the output of sensor measurements; here, the task is to find an operator that injects the output into the dynamics of the observer. In this talk, we discuss recent developments on the design of such feedback-input  and output-injection operators for models given by parabolic-like equations. The focus is put on the design of simple and explicit operators. Both theoretical and numerical aspects are discussed, including a comparison to more classical operators obtained through optimal control tools and involving the solution of Riccati or Hamilton-Jacobi-Bellman equations.