Related papers: The Polyhedral Geometry of Pivot Rules and Monoton…
In this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable…
To solve a linear program, the simplex method follows a path in the graph of a polytope, on which a linear function increases. The length of this path is an key measure of the complexity of the simplex method. Numerous previous articles…
If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex…
We show that the shadow vertex simplex algorithm can be used to solve linear programs in strongly polynomial time with respect to the number $n$ of variables, the number $m$ of constraints, and $1/\delta$, where $\delta$ is a parameter that…
A notion of "radially monotone" cut paths is introduced as an effective choice for finding a non-overlapping edge-unfolding of a convex polyhedron. These paths have the property that the two sides of the cut avoid overlap locally as the cut…
Estimating the number of vertices of a two dimensional projection, called a shadow, of a polytope is a fundamental tool for understanding the performance of the shadow simplex method for linear programming among other applications. We prove…
We consider facet-Hamiltonian cycles of polytopes, defined as cycles in their skeleton such that every facet is visited exactly once. These cycles can be understood as optimal watchman routes that guard the facets of a polytope. We consider…
Circuits play a fundamental role in the theory of linear programming due to their intimate connection to algorithms of combinatorial optimization and the efficiency of the simplex method. We are interested in better understanding the…
We investigate the worst-case behavior of the simplex algorithm on linear programs with three variables, that is, on 3-dimensional simple polytopes. Among the pivot rules that we consider, the ``random edge'' rule yields the best asymptotic…
We consider the computational problem of finding short paths in the skeleton of the perfect matching polytope of a bipartite graph. We prove that unless $P=NP$, there is no polynomial-time algorithm that computes a path of constant length…
We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan the least-squares estimate is given by a convex quadratic program.…
In optimal transport, quadratic regularization is a sparse alternative to entropic regularization: the solution measure tends to have small support. Computational experience suggests that the support decreases monotonically to the…
We prove that computing a shortest monotone path to the optimum of a linear program over a simple polytope is NP-hard, thus resolving a 2022 open question of De Loera, Kafer, and Sanit\`a. As a consequence, finding a shortest sequence of…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
Two methods are proposed for high-dimensional shape-constrained regression and classification. These methods reshape pre-trained prediction rules to satisfy shape constraints like monotonicity and convexity. The first method can be applied…
Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the…
In this paper, we analyze the simplex method with the largest distance rule and derive upper bounds on the number of different basic feasible solutions generated. The pivoting rule was proposed by Pan [10], and in some cases, it was…
Base polytopes of polymatroids, also known as generalized permutohedra, are polytopes whose edges are parallel to a vector of the form $\mathbf{e}_i - \mathbf{e}_j$. We consider the following computational problem: Given two vertices of a…
Pivoting methods are of vital importance for linear programming, the simplex method being the by far most well-known. In this paper, a primal-dual pair of linear programs in canonical form is considered. We show that there exists a sequence…
The Hirsch Conjecture stated that any $d$-dimensional polytope with n facets has a diameter at most equal to $n - d$. This conjecture was disproved by Santos (A counterexample to the Hirsch Conjecture, Annals of Mathematics, 172(1) 383-412,…