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Tropical polyhedra seem to play a central role in static analysis of softwares. These tropical geometrical objects play also a central role in parity games especially mean payoff games and energy games. And determining if an initial state…
We study the natural extended-variable formulation for the disjunction of $n+1$ polytopes in $\mathbb{R}^d$. We demonstrate that the convex hull $D$ in the natural extended-variable space $\mathbb{R}^{d+n}$ is given by full optimal big-M…
This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be…
Although deep learning has demonstrated remarkable capability in learning from unstructured data, modern tree-based ensemble models remain superior in extracting relevant information and learning from structured datasets. While several…
A set-system $S\subseteq \{0,1\}^n$ is cube-ideal if its convex hull can be described by capacity and generalized set covering inequalities. In this paper, we use combinatorics, convex geometry, and polyhedral theory to give exponential…
The hull of a linear code is defined as the intersection of the code and its dual. This concept was initially introduced to classify finite projective planes. The hull plays a crucial role in determining the complexity of algorithms used to…
Polynomial optimization problems are infinite-dimensional, nonconvex, NP-hard, and are often handled in practice with the moment-sums of squares hierarchy of semidefinite programming bounds. We consider problems where the objective function…
For a linear equality constrained convex optimization problem involving two objective functions with a ``nonsmooth" + ``nonsmooth" composite structure, we study two algorithms derived from a mixed-order dynamical system which incorporates…
We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in (Kaibel and Pfetsch, 2008). These polytopes are the convex hulls of all 0/1-matrices with…
Recent advances in cutting-plane strategies applied to robust optimization problems show that they are competitive with respect to problem reformulations and interior-point algorithms. However, although its application with polyhedral…
Decision trees are important both as interpretable models amenable to high-stakes decision-making, and as building blocks of ensemble methods such as random forests and gradient boosting. Their statistical properties, however, are not well…
We present a new universal source code for distributions of unlabeled binary and ordinal trees that achieves optimal compression to within lower order terms for all tree sources covered by existing universal codes. At the same time, it…
Recombining trinomial trees are a workhorse for modeling discrete-event systems in option pricing, logistics, and feedback control. Because each node stores a state-dependent quantity, a depth-$D$ tree naively yields $\mathcal{O}(3^{D})$…
With sound unification, Definite Clause Grammars and compact expression of combinatorial generation algorithms, logic programming is shown to conveniently host a declarative playground where interesting properties and behaviors emerge from…
We consider the problem of learning decision rules for prediction with feature budget constraint. In particular, we are interested in pruning an ensemble of decision trees to reduce expected feature cost while maintaining high prediction…
We introduce compositional tensor trains (CTTs) for the approximation of multivariate functions, a class of models obtained by composing low-rank functions in the tensor-train format. This format can encode standard approximation tools,…
Combinatorial evolution - the creation of new things through the combination of existing things - can be a powerful way to evolve rather than design technical objects such as electronic circuits. Intriguingly, this seems to be an ongoing…
In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete…
Polynomial, or Delsarte's, method in coding theory accounts for a variety of structural results on, and bounds on the size of, extremal configurations (codes and designs) in various metric spaces. In recent works of the authors the…
A tremendous range of design tasks in materials, physics, and biology can be formulated as finding the optimum of an objective function depending on many parameters without knowing its closed-form expression or the derivative. Traditional…