We study the value function of an integer program (IP) by characterizing how its optimal value changes as the right-hand side varies. We show that the IP value function can be approximated to any desired degree of accuracy using machine learning (ML) techniques. Since an IP value function is a Chvátal-Gomory (CG) function, we first … Read more
Bilevel optimization deals with nested problems in which a leader takes the first decision to minimize their objective function while accounting for a follower best-response reaction. Constrained bilevel problems with integer variables are particularly notorious for their hardness. While exact solvers have been proposed for mixed-integer~linear bilevel optimization, they tend to scale poorly with problem … Read more
The Fleet Sizing Problem (FSP) stands as a critical challenge within the realm of logistics and supply chain management, particularly in the context of Closed-Loop Supply Chains (CLSC). The significance of addressing the FSP lies in its direct impact on operational costs, resource utilization, and environmental sustainability. By effectively optimizing fleet size, organizations can streamline … Read more
We consider the problem of distributed task assignment in a swarm of Unmanned Aerial Vehicles (UAVs), where heterogeneous agents that can have different capabilities need to work in teams to execute tasks. We consider the case where communication between UAVs is costly or dangerous and should be limited or avoided, while individual UAVs have uncertain … Read more
It is well-known that coupling constraints in linear bilevel optimization can lead to disconnected feasible sets, which is not possible without coupling constraints. However, there is no difference between linear bilevel problems with and without coupling constraints w.r.t. their complexity-theoretical hardness. In this note, we prove that, although there is a clear difference between these … Read more
Partial inverse combinatorial optimization problems are bilevel optimization problems in which the leader aims to incentivize the follower to include a given set of elements in the solution of their combinatorial problem. If the set of required elements defines a complete follower solution, the inverse combinatorial problem is solvable in polynomial time as soon as … Read more
The length-constrained cycle partition problem (LCCP) is a graph optimization problem in which a set of nodes must be partitioned into a minimum number of cycles. Every node is associated with a critical time and the length of every cycle must not exceed the critical time of any node in the cycle. We formulate LCCP … Read more
Besides training, mathematical optimization is also used in deep learning to model and solve formulations over trained neural networks for purposes such as verification, compression, and optimization with learned constraints. However, solving these formulations soon becomes difficult as the network size grows due to the weak linear relaxation and dense constraint matrix. We have seen … Read more
We tackle the problems of workforce sizing and shift scheduling of a logistic operator delivering parcels in the last-mile segment of the supply chain. Our working hypothesis is that the relevant decisions are affected by two main trade-offs: workforce size and shift stability. A large workforce is able to deal with demand fluctuations but incurs … Read more
Decision trees are one of the most famous methods for solving classification problems, mainly because of their good interpretability properties. Moreover, due to advances in recent years in mixed-integer optimization, several models have been proposed to formulate the problem of computing optimal classification trees. The goal is, given a set of labeled points, to split … Read more