On the integrality Gap of Small Asymmetric Travelling Salesman Problems: A Polyhedral and Computational Approach

\(\) In this paper, we investigate the integrality gap of the Asymmetric Traveling Salesman Problem (ATSP) with respect to the linear relaxation given by the Asymmetric Subtour Elimination Problem (ASEP) for small-sized instances. In particular, we focus on the geometric properties and symmetries of the ASEP polytope ( \(P_{ASEP}^n\) ) and its vertices. The polytope’s … Read more

A Polyhedral Characterization of Linearizable Quadratic Combinatorial Optimization Problems

We introduce a polyhedral framework for characterizing instances of quadratic combinatorial optimization programs (QCOPs) that are linearizable, meaning that the quadratic objective can be equivalently rewritten as linear in such a manner that preserves the objective function value at all feasible solutions. In particular, we show that an instance is linearizable if and only if … Read more

The MIP Workshop 2023 Computational Competition on Reoptimization

This paper describes the computational challenge developed for a computational competition held in 2023 for the 20th anniversary of the Mixed Integer Programming Workshop. The topic of this competition was reoptimization, also known as warm starting, of mixed integer linear optimization problems after slight changes to the input data for a common formulation. The challenge … Read more

Adjustable robust optimization for fleet sizing problem in closed-loop supply chains with simultaneous delivery and pickup

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

Fourth-order Marginal Moment Model: Reformulations and Applications

This paper investigates the bounds on the expectation of combinatorial optimization given moment information for each individual random variable. A popular approach to solving this problem, known as the marginal moment model (MMM), is to reformulate it as a semidefinite program (SDP). In this paper, we investigate the structure of MMM with up to fourth-order … Read more

Solution methods for partial inverse combinatorial optimization problems in which weights can only be increased

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

Cuts and semidefinite liftings for the complex cut polytope

\(\) We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices \(xx^{\mathrm{H}}\), where the elements of \(x \in \mathbb{C}^n\) are \(m\)th unit roots. These polytopes have applications in \({\text{MAX-3-CUT}}\), digital communication technology, angular synchronization and more generally, complex quadratic programming. For \({m=2}\), the complex cut polytope corresponds to the well-known cut … Read more

On the power of linear programming for K-means clustering

In a previous work, we introduced a new linear programming (LP) relaxation for K-means clustering. In this paper, we further investigate the theoretical properties of this relaxation. We focus on K-means clustering with two clusters, which is an NP-hard problem. As evident from our numerical experiments with both synthetic and real-world data sets, the proposed … Read more

Bounds and Heuristic Algorithms for the Bin Packing Problem with Minimum Color Fragmentation

In this paper, we consider a recently introduced packing problem in which a given set of weighted items with colors has to be packed into a set of identical bins, while respecting capacity constraints and minimizing the total number of times that colors appear in the bins. We review exact methods from the literature and … Read more

Optimization Over Trained Neural Networks: Taking a Relaxing Walk

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