Vertices Of Polytope Research Articles

On the number of vertices of the stochastic tensor polytope

This paper studies lower and upper bounds for the number of vertices of the polytope of stochastic tensors (i.e. triply stochastic arrays of dimension n). By using known results on polytopes (i.e. the Upper and Lower Bound Theorems), we present some new lower and upper bounds. We show that the new upper bound is tighter than the one recently obtained by Chang et al. [Ann Funct Anal. 2016;7(3):386–393] and also sharper than the one in Linial and Luria’s [Discrete Comput Geom. 2014;51(1);161–170]. We demonstrate that the analog of the lower bound obtained in such a way, however, is no better than the existing ones.

Vertices of FFLV polytopes

FFLV polytopes describe monomial bases in irreducible representations of \(\mathfrak {sl}_n\) and \(\mathfrak {sp}_{2n}\). We study various sets of vertices of FFLV polytopes. First, we consider the special linear case. We prove the locality of the set of vertices with respect to the type A Dynkin diagram. Then we describe all the permutation vertices and after that we describe all the simple vertices and prove that their number is equal to the large Schröder number. Finally, we derive analogous results for symplectic Lie algebras.

4D Pyritohedral Symmetry

We describe an extension of the pyritohedral symmetry in 3D to 4-dimensional Euclidean space and construct the group elements of the 4D pyritohedral group of order 576 in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W (F4) and W (H4), implying that it is a group relevant to the crystallographic as well as quasicrystallographic structures in 4-dimensions. We derive the vertices of the 24 pseudoicosahedra, 24 tetrahedra and the 96 triangular pyramids forming the facets of the pseudo snub 24-cell. It turns out that the relevant lattice is the root lattice of W (D4). The vertices of the dual polytope of the pseudo snub 24-cell consists of the union of three sets: 24-cell, another 24-cell and a new pseudo snub 24-cell. We also derive a new representation for the symmetry group of the pseudo snub 24-cell and the corresponding vertices of the polytopes.

Power coordinates

We present a full geometric parameterization of generalized barycentric coordinates on convex polytopes. We show that these continuous and non-negative coefficients ensuring linear precision can be efficiently and exactly computed through a power diagram of the polytope's vertices and the evaluation point. In particular, we point out that well-known explicit coordinates such as Wachspress, Discrete Harmonic, Voronoi, or Mean Value correspond to simple choices of power weights. We also present examples of new barycentric coordinates, and discuss possible extensions such as power coordinates for non-convex polygons and smooth shapes.

Observer Design for LPV Systems With Uncertain Measurements on Scheduling Variables: Application to an Electric Ground Vehicle

In this paper, we aim to study the observer design problem for polytopic linear-parameter-varying (LPV) systems with uncertain measurements on scheduling variables. Due to the uncertain measurements, the uncertainties are considered in the weighting factors. It is assumed that the vertices of polytope are the same when the measurements on scheduling variables are uncertain and perfect. Then, an LPV system with the uncertain weighting factors can be transferred to an LPV system with uncertainties. To deal with the uncertainties and unknown disturbance in the observer design problem, we propose a gain-scheduling sliding mode observer. Defining the estimation error as the state vector minus the estimated state vector, the estimation error dynamics is established. The sliding mode observer design method is developed based on analysis results of the established estimation error system. The proposed observer design method is then applied to an electric ground vehicle (EGV) in which the measurement of longitudinal velocity is assumed to be uncertain. Experimental tests and comparisons are given to show the advantages of the proposed design method and the designed observer.

A non-negative representation learning algorithm for selecting neighbors

A crucial step in many manifold clustering and embedding algorithms involves selecting the neighbors of a data point. In this paper, a non-negative representation learning algorithm is proposed to select the neighbors of a data point, so that they are almost always lying in the tangent space of the manifold at the given query point. This is very important to multi-manifold clustering and manifold embedding. Convex geometry theory states that a data point on a face of a convex polytope can be a convex combination of the vertices of the polytope, and that the non-zero combination scalars correspond to the vertices of the face the point lies on. The intent of this paper is to use points that that are near to a point as the vertices of a convex polytope, and select the vertices on the same face as this point to be the neighbors. Clearly, the point can be a convex combination of its neighbors. We can observe that these neighbors almost always lie in the tangent space of the manifold at the query point and can preserve the local manifold structure well. Based on this basic idea, we construct an objective function. Moreover, an augmented Lagrange multipliers method is proposed for solving it to derive a non-negative representation, and then the neighbors of a data point can be obtained. The result of our neighbor selection algorithm can be used by other clustering methods such as LEM or manifold embedding methods such as LLE. We demonstrate the effectiveness and efficiency of the proposed method through experiments on synthetic data and a real-world problem.

Co-design of event-triggered [formula omitted] control for discrete-time linear parameter-varying systems with network-induced delays

The additional threshold of a mixed event-triggering mechanism makes it impossible to guarantee the requirement of asymptotic stability. As a consequence, it is necessary to investigate uniform ultimate boundedness which is a mild stability property. In this paper, we address event-triggered H∞ control for discrete-time linear parameter-varying systems by jointly designing a mixed event-triggering mechanism and a set of state feedback controllers. In the frame of time-delay system, we propose a parameter-dependent sufficient condition such that the closed-loop system with mixed event-triggering mechanism is globally uniformly ultimately bounded with a minimized ultimate bound for a certain disturbance attenuation level. As an extensive study, such sufficient condition is proved to be in a sense of global asymptotic stability when the additional threshold tends to zero. To avoid the infinite dimensional optimization problem, the proposed parameter-dependent co-design condition is relaxed as a finite set of linear matrix inequalities at the polytope vertices, which can be numerically trackable and computationally efficient. Two numerical examples are provided to demonstrate the effectiveness of the proposed method.

Numerical reconstruction of convex polytopes from directional moments

We reconstruct an n-dimensional convex polytope from the knowledge of its directional moments. The directional moments are related to the projection of the polytope vertices on a particular direction. To extract the vertex coordinates from the moment information we combine established numerical algorithms such as generalized eigenvalue computation and linear interval interpolation. Numerical illustrations are given for the reconstruction of 2-d and 3-d convex polytopes.

On Lattice-Free Orbit Polytopes

Given a permutation group acting on coordinates of $\mathbb{R}^n$, we consider lattice-free polytopes that are the convex hull of an orbit of one integral vector. The vertices of such polytopes are called \emph{core points} and they play a key role in a recent approach to exploit symmetry in integer convex optimization problems. Here, naturally the question arises, for which groups the number of core points is finite up to translations by vectors fixed by the group. In this paper we consider transitive permutation groups and prove this type of finiteness for the $2$-homogeneous ones. We provide tools for practical computations of core points and obtain a complete list of representatives for all $2$-homogeneous groups up to degree twelve. For transitive groups that are not $2$-homogeneous we conjecture that there exist infinitely many core points up to translations by the all-ones-vector. We prove our conjecture for two large classes of groups: For imprimitive groups and groups that have an irrational invariant subspace.

Combinatorial structure and adjacency of vertices of polytope of b-factors

In the present paper in terms of the graph theory we describe the structure and vertices adjacency criterion of b-factors polyhedron. The special attention is paid to nonintegral vertices. Results of the present paper, in particular, generalize properties of nonintegral vertices of TSP polyhedron, give vertices adjacency criterion of a transportation polytope.

On Fractional Realizations of Graph Degree Sequences

We introduce fractional realizations of a graph degree sequence and a closely associated convex polytope. Simple graph realizations correspond to a subset of the vertices of this polytope; we characterize degree sequences for which each polytope vertex corresponds to a simple graph realization. These include the degree sequences of threshold and pseudo-split graphs, and we characterize their realizations both in terms of forbidden subgraphs and graph structure.

Polytopic best-mean H ∞ performance analysis

In Boyarski and Shaked [(2005), ‘Robust H ∞ Control Design for Best Mean Performance Over an Uncertain-parameters Box’, Systems and Control Letters, 54, 585–595], a novel best-mean approach to robust analysis and control over uncertain-parameters boxes was presented. This article extends the results of Boyarski and Shaked (2005) to convex uncertainty polytopes of arbitrary shape and arbitrary number of vertices, without an underlying parameters model. The article addresses robust, polytopic, probabilistic H ∞ analysis of linear systems and focuses on the performance distribution over the uncertainty region (rather than just on the performance bound, as is customary in robust control). It is assumed that all system instances over the uncertainty polytope may occur with equal probability; this uniform distribution assumption is common in robust statistical analysis and is known to be conservative. The proposed approach considers different disturbance attenuation levels (DALs) at the vertices of the uncertainty polytope. It is shown that, under the latter assumption, the mean DAL over the polytope is the algebraic average of the DALs at the polytope's vertices. Thus, the mean DAL over the polytope can be optimised by minimising the sum of the DALs at the vertices. The standard deviation of the DAL over the uncertainty polytope is also addressed, and a method to minimise this standard deviation (in order to enforce uniform performance over the polytope) is shown. The example utilises a state-feedback synthesis theorem presented in Boyarski and Shaked (2005). A Monte-Carlo analysis verifies correct statistics of the resulting closed-loop ‘pointwise’ H ∞-norms over the uncertainty region, and highlights the differences from a corresponding bound minimisation: a marginally higher bound, but much better mean and variance.

Perturbation of transportation polytopes

We describe a perturbation method that can be used to reduce the problem of finding the multivariate generating function (MGF) of a non-simple polytope to computing the MGF of simple polytopes. We then construct a perturbation that works for any transportation polytope. We apply this perturbation to the family of central transportation polytopes of order kn×n, and obtain formulas for the MGFs of the feasible cone of each vertex of the polytope and the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order kn×n.

Computing tropical linear spaces

We define and study the cyclic Bergman fan of a matroid M, which is a simplicial polyhedral fan supported on the tropical linear space T(M) of M and is amenable to computational purposes. It slightly refines the nested set structure on T(M), and its rays are in bijection with flats of M which are either cyclic flats or singletons. We give a fast algorithm for calculating it, making some computational applications of tropical geometry now viable. Our C++ implementation, called TropLi, and a tool for computing vertices of Newton polytopes of A-discriminants, are both available online.

On the Structure of the Capacity Region of Asynchronous Memoryless Multiple-Access Channels

The asynchronous capacity region of memoryless multiple-access channels is the union of certain polytopes. It is well-known that vertices of such polytopes may be approached via a technique called successive decoding. It is also known that an extension of successive decoding applies to the dominant face of such polytopes. The extension consists of forming groups of users in such a way that users within a group are decoded jointly whereas groups are decoded successively. This paper goes one step further. It is shown that successive decoding extends to every face of the above mentioned polytopes. The group composition as well as the decoding order for all rates on a face of interest are obtained from a label assigned to that face. From the label one can extract a number of structural properties, such as the dimension of the corresponding face and whether or not two faces intersect. Expressions for the the number of faces of any given dimension are also derived from the labels.

Perturbation of transportation polytopes

We describe a perturbation method that can be used to compute the multivariate generating function (MGF) of a non-simple polyhedron, and then construct a perturbation that works for any transportation polytope. Applying this perturbation to the family of central transportation polytopes of order $kn \times n$, we obtain formulas for the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order $kn \times n$. Nous décrivons une méthode de perturbation qui peut être utilisée pour calculer la fonction génératrice multivariée (MGF) d'un polyèdre non-simple, et ensuite construire une perturbation qui fonctionne pour tout polytope de transport. Appliquant cette perturbation à la famille des centraux de transport polytopes de l'ordre $kn \times n$, nous obtenons des formules pour le MGF du polytope. Les formules que nous obtenons sont énumérées par les objets combinatoires. Un cas spécial des formules récupère les résultats sur des polytopes de Birkhoff donnés par l'auteur et De Loera et Yoshida. Nous récupérons également la formule pour le nombre de sommets maximum des de transport polytopes d'ordre $kn \times n$.

An ellipsoidal off-line robust model predictive control strategy for uncertain polytopic discrete-time systems

In this paper, an off-line robust model predictive control strategy for uncertain polytopic discrete-time systems is presented. The on-line computational burdens are reduced by precomputing off-line the sequence of state feedback gains corresponding to the sequences of nested ellipsoids. The number of sequences of nested ellipsoids constructed is equal to the number of vertices of polytope. At each sampling instant, the smallest ellipsoid containing the currently measured state is determined in each sequence of ellipsoids. The real-time state feedback gain is then calculated by solving a numerically low-demanding optimization problem. The nominal model of the plant is included in the controller design in order to improve control performance. An overall algorithm is proved to guarantee robust stability.

On the stability and control of discrete linear systems with clock synchronisation errors

This article considers discrete linear time-invariant systems that can be decomposed into subsystems whose states are synchronised by a common clock with a signal that reaches them with delays. In particular, stability for the case where all subsystems have the same sampling frequency, but different switching times, is investigated. In contrast to previous work, the approach taken here models the set of system matrices that arise using a polytopic uncertainty description, which has seen extensive application in robust control theory for linear systems. Stabilisation is then achieved by state feedback and a method to handle the combinatorial explosion of the number of polytope vertices is developed and illustrated using an example from swarm system navigation.

REMARKS ON SELF-AFFINE FRACTALS WITH POLYTOPE CONVEX HULLS

Suppose that the set [Formula: see text] of real n × n matrices has joint spectral radius less than 1. Then for any digit set D = {d1, …, dq} ⊂ ℝn, there exists a unique non-empty compact set [Formula: see text] satisfying [Formula: see text], which is typically a fractal set. We use the infinite digit expansions of the points of F to give simple necessary and sufficient conditions for the convex hull of F to be a polytope. Additionally, we present a technique to determine the vertices of such polytopes. These answer some of the related questions of Strichartz and Wang, and also enable us to approximate the Lebesgue measure of such self-affine sets. To show the use of our results, we also give several examples including the Levy dragon and the Heighway dragon.

Converting between quadrilateral and standard solution sets in normal surface theory

The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3-manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson's Q-theory speeds up this operation by using a much smaller space (quadrilateral coordinates), at the cost of a reduced solution set that might not always be sufficient for our needs. In this paper we present algorithms for converting between solution sets in quadrilateral and standard coordinates. As a consequence we obtain a new algorithm for enumerating all standard vertex normal surfaces, yielding both the speed of quadrilateral coordinates and the wider applicability of standard coordinates. Experimentation with the software package Regina shows this new algorithm to be extremely fast in practice, improving speed for large cases by factors from thousands up to millions.