Related papers: Convex Hulls, Oracles, and Homology
Polynomial systems occur in many fields of science and engineering. Polynomial homotopy continuation methods apply symbolic-numeric algorithms to solve polynomial systems. We describe the design and implementation of our web interface and…
In this paper we look for the convex hull of a set using the geometric evolution by minimal curvature of a hypersurface that surrounds the set. To find the convex hull, we study the large time behavior of solutions to an obstacle problem…
Let $P$ be a planar set of $n$ points in general position. We consider the problem of computing an orientation of the plane for which the Rectilinear Convex Hull of $P$ has minimum area. Bae et al. (Computational Geometry: Theory and…
We describe a new approach to certifying the global nonnegativity of multivariate polynomials by solving hyperbolic optimization problems---a class of convex optimization problems that generalize semidefinite programs. We show how to…
We describe, study, and experiment with an algorithm for finding all solutions of systems of polynomial equations using homotopy continuation and monodromy. This algorithm follows a framework developed in previous work and can operate in…
We say that a subset of C^n is hypoconvex if its complement is the union of complex hyperplanes. Let D be the closed unit disk in C, T the unit circle. We prove two conjectures of Helton and Marshall. (See ``Frequency domain design and…
Constant workspace algorithms use a constant number of words in addition to the read-only input to the algorithm. In this paper, we devise algorithms to efficiently compute relative hulls in the plane using a constant workspace.…
The central problem in computational algebraic topology is the computation of the homotopy groups of a given space, represented as a simplicial set. Algorithms have been found which achieve this, but the running times depend on the size of…
In this paper we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and…
We propose a symmetric key homomorphic encryption scheme based on the evaluation of multivariate polynomials over a finite field. The proposed scheme is somewhat homomorphic with respect to addition and multiplication. Further, we define a…
Convexification is a core technique in global polynomial optimization. Currently, there are two main approaches competing in theory and practice: the approach of nonlinear programming and the approach based on positivity certificates from…
This paper is about solving polynomial systems. It first recalls how to do that efficiently with a very high probability of correctness by reconstructing a rational univariate representation (rur) using Groebner revlex computation,…
We propose a conservative algorithm to test the geometrical validity of simplicial (triangles, tetrahedra), tensor product (quadrilaterals, hexahedra), and mixed (prisms) elements of arbitrary polynomial order as they deform over a…
Recent research has examined algorithms to minimize robots' resource footprints. The class of combinatorial filters (discrete variants of widely-used probabilistic estimators) has been studied and methods for reducing their space…
Binary polynomial optimization is equivalent to the problem of minimizing a linear function over the intersection of the multilinear set with a polyhedron. Many families of valid inequalities for the multilinear set are available in the…
This paper deals with the regularization of the sum of functions defined on a locally convex spaces through their closed-convex hulls in the bidual space. Different conditions guaranteeing that the closed-convex hull of the sum is the sum…
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for…
The matrix code equivalence problem consists, given two matrix spaces $\mathcal{C},\mathcal{D} \subset \mathbb{F}_q^{m\times n}$ of dimension $k$, in finding invertible matrices $P\in\mathrm{GL}_m(\mathbb{F}_q)$ and…
Convex hulls are fundamental geometric tools used in a number of algorithms. This paper presents a fast, simple to implement and robust Smart Convex Hull (S-CH) algorithm for computing the convex hull of a set of points in E3. This…