The classical Unit Commitment problem is a MILP that can optimize thermal unit schedules to minimize costs and/or emissions while meeting demand and operational constraints inside and in-between stages (e.g. minimum uptime constraints). Integrating renewable energy introduces intermittency, leading to Uncertain Unit Commitment (UUC) models that necessitate efficient computational methods. Our collaboration with TotalEnergies focuses on an isolated industrial microgrid with renewable, thermal, and energy storage resources.
Addressing uncertainty involves encoding risk attitudes, with approaches ranging from risk neutral—minimizing the expected objective—to robust—minimizing worst-case—, the latter of which is our focus. Furthermore, information models dictate the dependency of decisions on random events. As an approximation, we investigate models where different variables have different dependencies. In our proposal, binary decisions over the entire horizon are made at the first time step and continuous decisions are made as they need to be executed, ensuring tractability and feasibility.
Efficiently solving UUC problems demands specialized techniques. We propose cut-based decomposition methods, including adaptations of Stochastic and Robust Dual Dynamic Programming algorithms. We assess algorithmic performance and solution quality through numerical experiments.

Distributionally robust optimization (DRO) is currently a hot topic in literature due to its ability to provide robust solutions in the face of uncertain parameters with unknown probability distributions. The core idea behind DRO is to find a solution that remains resilient even under the worst-case scenario by constructing an ambiguity set of probability distributions around the nominal distribution. This nominal distribution is typically estimated from sampled data.

Among the methods used to construct ambiguity sets, metric-based approaches are popular. These methods involve defining an ambiguity set comprising all candidate distributions that are within a certain distance from the nominal distribution, as prescribed by a chosen probability distance metric. The Wasserstein distance is a well-studied metric for DRO. However, the conventional formulation is only manageable under specific assumptions on the constraint functions and may lead to overly cautious solutions, ignoring any existing distributional characteristics of the samples, such as multimodality.

To overcome the limitations of traditional approaches, our work proposes a variant of the optimal transport problem between Gaussian mixture models (GMM) to construct the ambiguity set for DRO. In this variant, optimal transport is performed between discrete measures, with supports represented by the Gaussian components of the GMM. In our method, the nominal distribution is estimated as a GMM from the sampled data, which often exhibits multimodality in practical scenarios. We also derive a tractable formulation for the distributionally robust chance-constrained problem using the GMM-based ambiguity set, approximating the probabilistic constraints with Conditional Value-at-Risk (CVaR). Case studies demonstrate that our proposed model outperforms traditional Wasserstein DRO methods when dealing with multimodal distributed uncertainty.

The kidney Exchange Problem (KEP) can be defined on a directed weighted graph where the vertices represent the altruistic donors and the incompatible patient-donor pairs and the arcs represent the possible transplants.
The arc weights encode the medical benefit which is a measure of the quality of the associated transplant.
In such a graph, kidney exchanges can be modelled as circuits starting and ending in an incompatible pair or as paths starting at an altruistic donor.
Due to medical reasons, circuits and paths are subject to length constraints.
The aim of the KEP is to determine a set of disjoint kidney exchanges of maximal medical benefit or maximal cardinality (all weights equal to one).
In this work, we consider two types of uncertainty in the KEP which stem from the estimation of the medical benefit (weights of the arcs) and from the failure of a transplantation (existence of an arc).
We model both uncertainties by means of uncertainty sets with constant budget.
The robust approach entails finding the best KEP solution in the worst-case scenario within the uncertainty set.
We modelled the robust counterpart via a max-min formulation which is defined on exponentially-many variables associated with the circuits and paths.
We propose two exact approaches to solve it.
The first one exploits the well-known result of Bertsimas and Sim, whereas the second one is based on a reformulation of the robust counterpart to a single-level problem.
Both solution approaches rely on a Branch-Price-and-Cut algorithm where the exponentially-many variables are dynamically generated.
The computational experiments prove the efficiency of our approaches.