For almost two decades, mixed integer programming (MIP) solvers have used graph- based conflict analysis to learn from local infeasibilities during branch-and-bound search. In this paper, we improve MIP conflict analysis by instead using reasoning based on cuts, inspired by the development of conflict-driven solvers for pseudo- Boolean optimization. Phrased in MIP terminology, this type … Read more
For equality-constrained linear mixed-integer programs (MIP) defined by rational data, it is known that the subadditive dual is a strong dual and that there exists an optimal solution of a particular form, termed generator subadditive function. Motivated by these results, we explore the connection between Lagrangian duality, subadditive duality and generator subadditive functions for general … Read more
Datasets collected for analysis often contain a certain amount of incomplete instances, where some feature values are missing. Since many statistical analyses and machine learning algorithms depend on complete datasets, missing values need to be imputed in advance. Bertsimas et al. (2018) proposed a high-performance method that combines machine learning and mathematical optimization algorithms for … Read more
We present column elimination as a general framework for solving (large-scale) integer programming problems. In this framework, solutions are represented compactly as paths in a directed acyclic graph. Column elimination starts with a relaxed representation, that may contain infeasible paths, and solves a constrained network flow over the graph to find a solution. It then … Read more
In the quadratic minimum spanning tree problem (QMSTP) one wants to find the minimizer of a quadratic function over all possible spanning trees of a graph. We present a formulation of the QMSTP as a mixed-integer semidefinite program exploiting the algebraic connectivity of a graph. Based on this formulation, we derive a doubly nonnegative relaxation … Read more
While mixed-integer linear programming and convex programming solvers have advanced significantly over the past several decades, solution technologies for general mixed-integer nonlinear programs (MINLPs) have yet to reach the same level of maturity. Various problem structures across different application domains remain challenging to model and solve using modern global solvers, primarily due to the lack … Read more
Efficient operation of underground railway systems is critical not only for maintaining punctual service but also for minimizing energy consumption, a key factor in reducing operational costs and environmental impact. To evaluate the energy consumption of the timetables, this paper delves into the development of mathematical models to accurately represent energy dynamics within the underground … Read more
We report on results concerning the polyhedral structure of, and integer linear programming formulations for, the quadratic linear ordering problem. Specifically, we provide a deeper analysis of the characteristic equation system that takes part in the minimal description of the convex hull of its feasible solutions, and we determine an accessible description of a restricted … Read more
We introduce a new class of valid inequalities for any 0/1 knapsack polytope, called Solitary item inequality, which are facet-defining. We prove that any facet-defining inequality of a 0/1 knapsack polytope with nonnegative integral coefficients and right hand side 1 belongs to this class, and hence, the set of facet-defining inequalities corresponding to strong covers … Read more
In the Vehicle Routing Problem (VRP) with Lockers, the vertices in a graph are divided into two key subsets: a customer set and a locker set, with lockers serving as alternative delivery points for the customer’s parcel. This paper presents a generic VRP with lockers, where a customer’s parcel can be delivered either to its … Read more