Related papers: Computing convex hulls and counting integer points…

We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic…

We study approximations of polytopes in the standard model for computing polytopes using Minkowski sums and (convex hulls of) unions. Specifically, we study the ability to approximate a target polytope by polytopes of a given depth. Our…

The Fourier transforms of polyhedral cones can be used, via Brion's theorem, to compute various geometric quantities of polytopes, such as volumes, moments, and lattice-point counts. We present a novel method of computing these conic…

A new approach to the local and global explanation is proposed. It is based on selecting a convex hull constructed for the finite number of points around an explained instance. The convex hull allows us to consider a dual representation of…

We state the formula for the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given sublattice and give a partial proof of the formula. We show that the proof can be…

An incremental approach for computation of convex hull for data points in two-dimensions is presented. The algorithm is not output-sensitive and costs a time that is linear in the size of data points at input. Graham's scan is applied only…

In this paper, we explore the challenge of assortment planning in the context of quick-commerce, a rapidly-growing business model that aims to deliver time-sensitive products. In order to achieve quick delivery to satisfy the immediate…

We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal subject to a group acting on the columns. Special cases are packing and partitioning orbitopes, which arise from restrictions to matrices with at…

Taking the convex hull of a curve is a natural construction in computational geometry. On the other hand, path signatures, central in stochastic analysis, capture geometric properties of curves, although their exact interpretation for…

This article discusses the problem of determining whether a given point, or set of points, lies within the convex hull of another set of points in $d$ dimensions. This problem arises naturally in a statistical context when using a…

Let T be the unit circle in the complex plane C. This paper proves the existence of analytic structure in a compact subset K of T X C^n, where K has so-called "lineally convex" or "hypoconvex" fibers over T. It also addresses a related…

We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial…

The convex hull of a set of points, $C$, serves to expose extremal properties of $C$ and can help identify elements in $C$ of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be…

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…

The mixed volume counts the roots of generic sparse polynomial systems. Mixed cells are used to provide starting systems for homotopy algorithms that can find all those roots, and track no unnecessary path. Up to now, algorithms for that…

Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, we define several new families of polytopes as convex hulls of sign matrices, which are certain {0,1,-1}-matrices in…

We study the convex hull of the graph of a quadratic function $f(\mathbf{x})=\sum_{ij\in E}x_ix_j$, where the sum is over the edge set of a graph $G$ with vertex set $\{1,\dots,n\}$. Using an approach proposed by Gupte et al. (Discrete…

We study a correspondence between numerical sets and integer partitions that leads to a bijection between simultaneous core partitions and the integer points of a certain polytope. We use this correspondence to prove combinatorial results…

Linear codes play a central role in coding theory and have applications in several branches of mathematics. For error correction purposes the minimum Hamming distance should be as large as possible. Linear codes related to applications in…

We study the generalization of split, k-branch split, and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the…