We consider linear relaxations for multilinear optimization problems. In a recent paper, Khajavirad proved that the extended flower relaxation is at least as strong as the relaxation of any recursive McCormick linearization (Operations Research Letters 51 (2023) 146-152). In this paper we extend the result to more general linearizations, and present a simpler proof. Moreover, … Read more
Robust combinatorial optimization with budget uncertainty is one of the most popular approaches for integrating uncertainty into optimization problems. The existence of a compact reformulation for (mixed-integer) linear programs and positive complexity results give the impression that these problems are relatively easy to solve. However, the practical performance of the reformulation is quite poor when … Read more
In the past decade, deep learning became the prevalent methodology for predictive modeling thanks to the remarkable accuracy of deep neural networks in tasks such as computer vision and natural language processing. Meanwhile, the structure of neural networks converged back to simpler representations based on piecewise constant and piecewise linear functions such as the Rectified … Read more
Let $A\in \mathbb{R}^{m\times n}\setminus \{0\}$ and $P:=\{x:Ax\le 0\}$. This paper provides a procedure to compute an upper bound on the following {\em homogeneous Hoffman constant} \[ H_0(A) := \sup_{u\in \mathbb{R}^n \setminus P} \frac{\text{dist}(u,P)}{\text{dist}(Au, \mathbb{R}^m_-)}. \] In sharp contrast to the intractability of computing more general Hoffman constants, the procedure described in this paper is entirely … Read more
In this paper we study the Hamiltonian \(p\)-median problem, in which a weighted graph on \(n\) vertices is to be partitioned into \(p\) simple cycles of minimum total weight. We introduce two new families of valid inequalities for a formulation of the problem in the space of natural edge variables. Each one of the families … Read more
Decision trees are a widely used tool for interpretable machine learning. Multivariate decision trees employ hyperplanes at the branch nodes to route datapoints throughout the tree and yield more compact models than univariate trees. Recently, mixed-integer programming (MIP) has been applied to formulate the optimal decision tree problem. To strengthen MIP formulations, it is crucial … Read more
DR-submodular functions encompass a broad class of functions which are generally non-convex and non-concave. We study the problem of minimizing any DR-submodular function, with continuous and general integer variables, under box constraints and possibly additional monotonicity constraints. We propose valid linear inequalities for the epigraph of any DR-submodular function under the constraints. We further provide … Read more
For a subset T of nodes of an undirected graph G, a T-Steiner cut is a cut δ(S) where S intersects both T and the complement of T. The T-Steiner cut dominant} of G is the dominant CUT_+(G,T) of the convex hull of the incidence vectors of the T-Steiner cuts of G. For T={s,t}, this … Read more
Motivated by applications in cloud computing, we study a temporal bin packing problem with jobs that occupy half of a bin’s capacity. An instance is given by a set of jobs, each with a start and end time during which it must be processed, i.e., assigned to a bin. A bin can accommodate two jobs … Read more
We investigate the problem of optimizing a linear objective function over the set of all binary vectors of length n with bounded variation, where the latter is defined as the number of pairs of consecutive entries with different value. This problem arises naturally in many applications, e.g., in unit commitment problems or when discretizing binary … Read more