Andrea Lodi In recent years, the integration of Machine Learning (ML) models with Operation Research (OR) tools has gained popularity across diverse applications, including cancer treatment, algorithmic configuration, and chemical process optimization. In this domain, the combination of ML and OR often relies on representing the ML model output using Mixed Integer Programming (MIP) formulations. Numerous studies … Read more
Dantzig-Wolfe (DW) decomposition is a well-known technique in mixed-integer programming (MIP) for decomposing and convexifying constraints to obtain potentially strong dual bounds. We investigate cutting planes that can be derived using the DW decomposition algorithm and show that these cuts can provide the same dual bounds as DW decomposition. More precisely, we generate one cut … Read more
Motivated by recent progress on online linear programming (OLP), we study the online decision making problem (ODMP) as a natural generalization of OLP. In ODMP, there exists a single decision maker who makes a series of decisions spread out over a total of \(T\) time stages. At each time stage, the decision maker makes a … Read more
Bike-sharing systems have been implemented in multiple major cities, offering a low-cost and environmentally friendly transportation alternative to vehicles. Due to the stochastic nature of customer trips, stations are often unbalanced, resulting in unsatisfied demand. As a remedy, operators employ trucks to rebalance bikes among unbalanced stations. Given the complexity of the dynamic rebalancing planning, … Read more
Finding high-quality solutions to mixed-integer linear programming problems (MILPs) is of great importance for many practical applications. In this respect, the refinement heuristic local branching (LB) has been proposed to produce improving solutions and has been highly influential for the development of local search methods in MILP. The algorithm iteratively explores a sequence of solution … Read more
We study a class of chance-constrained two-stage stochastic optimization problems where the second-stage recourse decisions belong to mixed-integer convex sets. Due to the nonconvexity of the second-stage feasible sets, standard decomposition approaches cannot be applied. We develop a provably convergent branch-and-cut scheme that iteratively generates valid inequalities for the convex hull of the second-stage feasible … Read more
We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint on the general problem classes and models, and give concrete examples from diverse application fields such as signal and image processing, portfolio selection, or machine learning. The paper discusses general-purpose modeling techniques and … Read more
Quadratically-constrained quadratic programs (QCQPs) are optimization models whose remarkable expressiveness has made them a cornerstone of methodological research for nonconvex optimization problems. However, modern methods to solve a general QCQP fail to scale, encountering computational challenges even with just a few hundred variables. Specifically, a semidefinite programming (SDP) relaxation is typically employed, which provides strong … Read more
Decision making problems are typically concerned with maximizing efficiency. In contrast, we address problems where there are multiple stakeholders and a centralized decision maker who is obliged to decide in a fair manner. Different decisions give different utility to each stakeholder. In cases where these decisions are made repeatedly, we provide efficient mathematical programming formulations … Read more
The concept of balance plays an important role in many combinatorial optimization problems. Yet there exist various ways of expressing balance, and it is not always obvious how best to achieve it. In this methodology-focused paper, we study three cases where its integration is deficient and analyze the causes of these inadequacies. We examine the … Read more