We report on results concerning the polyhedral structure of the quadratic linear ordering problem and its associated integer linear programming formulations. Specifically, we provide a deeper analysis of the characteristic equation system that partly describes the corresponding polytope, i.e., the convex hull of the feasible solutions to the quadratic linear ordering problem, and determine an … 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
This tutorial provides an introduction to the use of decision diagrams for solving discrete optimization problems. A decision diagram is a graphical representation of the solution space, representing decisions sequentially as paths from a root node to a target node. By merging isomorphic subgraphs (or equivalent subproblems), decision diagrams can compactly represent an exponential solution … Read more
The most successful approaches for the TSP use the integer programming model proposed in 1954 by Dantzig, Fulkerson, and Johnson (DFJ). Although this model has exponentially many subtour elimination constraints (SECs), it has been observed that relatively few of them are needed to prove optimality in practice. This leads us to wonder: What is the … Read more
The maximum-entropy sampling problem (MESP) aims to select the most informative principal submatrix of a prespecified size from a given covariance matrix. This paper proposes an augmented factorization bound for MESP based on concave relaxation. By leveraging majorization and Schur-concavity theory, we demonstrate that this new bound dominates the classic factorization bound of Nikolov (2015) and a recent … Read more
This paper is devoted to the Truck-to-dock Door Assignment Problem. Two integer programming formulations introduced after 2009 are examined. Our review of the literature takes note of the criticisms and limitations addressed to the seminal work of 2009. Although the published adjustments that followed present strong argument and technical background, we have identified several errors, … Read more
We introduce the prime programming problem as a subclass of integer programming. These optimization models impose the restriction of feasible solutions being prime numbers. Then, we demonstrate how several classical problems in number theory can be formulated as prime programs. To solve such problems with a commercial optimization solver, we extend the branch-and-bound procedure of … Read more
We present randomized reconstruction approaches for optimal solutions to mixed-integer elliptic PDE control systems. Approximation properties and relations to sum-up rounding are derived using the cut norm. This enables us to dispose of space-filling curves required for sum-up rounding. Rates of almost sure convergence in the cut norm and the SUR norm in control space … Read more
We are excited to present the early release of Part I of our book “Optimizing with Column Generation: advanced Branch-Cut-and-Price Algorithms”. While the book’s ultimate goal, as suggested by its subtitle, is to describe cutting-edge techniques in these algorithms, this objective is primarily addressed in the forthcoming Part II. However, we feel that the completed … Read more