It is common knowledge that symmetries arising in integer programs could impair the solution process, in particular when symmetric solutions lead to an excessively large branch and bound (B&B) search tree. Techniques like isomorphic pruning [11], orbital branching [16] and orbitopal fixing [8] have been shown to be essential to solve very symmetric instances from … Read more
In this paper we investigate non-linear matroid optimization problems with polynomial objective functions where the monomials satisfy certain monotonicity properties. Indeed, we study problems where the set of non-linear monomials consists of all non-linear monomials that can be built from a given subset of the variables. Linearizing all non-linear monomials we study the respective polytope. … Read more
Given two disjoint convex polyhedra, we look for a best approximation pair relative to them, i.e., a pair of points, one in each polyhedron, attaining the minimum distance between the sets. Cheney and Goldstein showed that alternating projections onto the two sets, starting from an arbitrary point, generate a sequence whose two interlaced subsequences converge … Read more
The matroid parity (MP) problem is a powerful (and NP-hard) extension of the matching problem. Whereas matching polytopes are well understood, little is known about MP polytopes. We prove that, when the matroid is laminar, the MP polytope is affinely congruent to a perfect b-matching polytope. From this we deduce that, even when the matroid … Read more
In the study of extensions of polytopes of combinatorial optimization problems, a notorious open question is that for the size of the smallest extended formulation of the Minimum Spanning Tree problem on a complete graph with n nodes. The best known lower bound is \Omega(n^2), the best known upper bound is O(n^3). In this note … Read more
A fooling-set matrix is a square matrix with nonzero diagonal, but at least one in every pair of diagonally opposite entries is 0. Dietzfelbinger et al. ’96 proved that the rank of such a matrix is at least $\sqrt n$, for a matrix of order $n$. We ask for the typical minimum rank of a … Read more
In this paper we investigate the family of functions representable by deep neural networks (DNN) with rectified linear units (ReLU). We give the first-ever polynomial time (in the size of data) algorithm to train a ReLU DNN with one hidden layer to {\em global optimality}. This follows from our complete characterization of the ReLU DNN … Read more
We construct two families of Dantzig figures, which are $d$-dimensional polytopes with $2d$ facets and an antipodal vertex pair, from convex hulls of initial subsets for the graded lexicographic (grlex) and graded reverse lexicographic (grevlex) orders on $\mathbb{Z}^{d}_{\geq 0}$. These polytopes have the same number of vertices $O(d^2)$ and the same number of edges $O(d^3)$, … Read more
The Min-up/min-down Unit Commitment Problem (MUCP) is to find a minimum-cost production plan on a discrete time horizon for a set of fossil-fuel units for electricity production. At each time period, the total production has to meet a forecasted demand. Each unit must satisfy minimum up-time and down-time constraints besides featuring production and start-up costs. … Read more
In this paper we analyze general Stackelberg games (SGs) and Stackelberg security games (SSGs). SGs are hierarchical adversarial games where players select actions or strategies to optimize their payoffs in a sequential manner. SSGs are a type of SGs that arise in security applications, where the strategies of the player that acts first consist in … Read more