scholarly journals Set characterizations and convex extensions for geometric convex-hull proofs

2021 ◽  
Author(s):  
Andreas Bärmann ◽  
Oskar Schneider
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Expressive Power ◽  
Original Method ◽  
Proof Technique ◽  
Practical Usefulness ◽  
Algebraic Framework ◽  
Significant Extension ◽  
General Convex ◽  
Bilinear Functions

AbstractIn the present work, we consider Zuckerberg’s method for geometric convex-hull proofs introduced in Zuckerberg (Oper Res Lett 44(5):625–629, 2016). It has only been scarcely adopted in the literature so far, despite the great flexibility in designing algorithmic proofs for the completeness of polyhedral descriptions that it offers. We suspect that this is partly due to the rather heavy algebraic framework its original statement entails. This is why we present a much more lightweight and accessible approach to Zuckerberg’s proof technique, building on ideas from Gupte et al. (Discrete Optim 36:100569, 2020). We introduce the concept of set characterizations to replace the set-theoretic expressions needed in the original version and to facilitate the construction of algorithmic proof schemes. Along with this, we develop several different strategies to conduct Zuckerberg-type convex-hull proofs. Very importantly, we also show that our concept allows for a significant extension of Zuckerberg’s proof technique. While the original method was only applicable to 0/1-polytopes, our extended framework allows to treat arbitrary polyhedra and even general convex sets. We demonstrate this increase in expressive power by characterizing the convex hull of Boolean and bilinear functions over polytopal domains. All results are illustrated with indicative examples to underline the practical usefulness and wide applicability of our framework.

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

scholarly journals A Sufficient Condition for Convex Hull Property in General Convex Spatio-Temporal Corridors

2021 ◽  
Author(s):  
Weize Zhang ◽  
Peyman Yadmellat ◽  
Zhiwei Gao
Keyword(s):  
Convex Hull ◽  
Motion Planning ◽  
Autonomous Driving ◽  
Search Space ◽  
Sufficient Condition ◽  
Simple Shape ◽  
Planning Approach ◽  
General Convex ◽  
Simulation Results ◽  
Spatio Temporal

Motion planning is one of the key modules in autonomous driving systems to generate trajectories for self-driving vehicles to follow. A common motion planning approach is to generate trajectories within semantic safe corridors. The trajectories are generated by optimizing parametric curves (e.g. Bezier curves) according to an objective function. To guarantee safety, the curves are required to satisfy the convex hull property, and be contained within the safety corridors. The convex hull property however does not necessary hold for time-dependent corridors, and depends on the shape of corridors. The existing approaches only support simple shape corridors, which is restrictive in real-world, complex scenarios. In this paper, we provide a sufficient condition for general convex, spatio-temporal corridors with theoretical proof of guaranteed convex hull property. The theorem allows for using more complicated shapes to generate spatio-temporal corridors and minimizing the uncovered search space to $O(\frac{1}{n^2})$ compared to $O(1)$ of trapezoidal corridors, which can improve the optimality of the solution. Simulation results show that using general convex corridors yields less harsh brakes, hence improving the overall smoothness of the resulting trajectories.

scholarly journals Progress in the development of the semi-empirical calibration of neutron probes based on Monte Carlo modeling

2021 ◽  
Vol 46 (3) ◽  
pp. 251
Author(s):  
Urszula Woźnicka
Keyword(s):  
Monte Carlo ◽  
Numerical Experiments ◽  
Original Method ◽  
Neutron Probe ◽  
The Monte Carlo Method ◽  
Empirical Calibration ◽  
Wide Range ◽  
Significant Extension ◽  
Simulation Results ◽  
Semi Empirical

The method of the semi-empirical calibration of a neutron well logging probe was developed by Jan Andrzej Czubek on the concept of the general neutron parameter (GNP) and tested positively at the neutron calibration station in Zielona Góra, Poland. The neutron probe responses in a wide range of neutron parameters (and thus lithology, porosity and saturation) were also computed using the Monte Carlo method. The obtained simulation results made it possible to determine the calibration curves using the Czubek concept in a wider range than by means of the original method. The very good compatibility of both methods confirms the applicability of the GNP as well as the Monte Carlo numerical experiments, which allow for a significant extension of the semi-empirical calibration in complex well geometries taking into account e.g., casing or invaded zones.

The distribution of the convex hull of a Gaussian sample

1980 ◽  
Vol 17 (03) ◽  
pp. 686-695 ◽  
Author(s):  
William F. Eddy
Keyword(s):  
Weak Convergence ◽  
Convex Hull ◽  
Poisson Point Process ◽  
Point Processes ◽  
Convex Sets ◽  
Continuous Mapping ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Limit Process ◽  
Gaussian Sample

The distribution of the convex hull of a random sample ofd-dimensional variables is described by embedding the collection of convex sets into the space of continuous functions on the unit sphere. Weak convergence of the normalized convex hull of a circular Gaussian sample to a process with extreme-value marginal distributions is demonstrated. The proof shows that an underlying sequence of point processes converges to a Poisson point process and then applies the continuous mapping theorem. Several properties of the limit process are determined.

Metrizability of Finite Dimensional Spaces with a Binary Convexity

1986 ◽  
Vol 38 (1) ◽  
pp. 1-18 ◽  
Author(s):  
M. Van de Vel
Keyword(s):  
Convex Hull ◽  
Weak Topology ◽  
Convex Sets ◽  
Nonempty Intersection ◽  
Finite Collection ◽  
Closed Sets ◽  
Finite Dimensional ◽  
The One ◽  
Image Position ◽  
Disjoint Convex

A convex structure consists of a set X, together with a collection of subsets of X, which is closed under intersection and under updirected union. The members of are called convex sets, and is a convexity on X. Fox A a subset of X, h (A) denotes the (convex) hull of A. If A is finite, then h(A) is called a polytope, is called a binary convexity if each finite collection of pairwise intersecting convex sets has a nonempty intersection. See [8], [21] for general references.If X is also equipped with a topology, then the corresponding weak topology is the one generated by the convex closed sets. It is usually assumed that at least all polytopes are closed. is called normal provided that for each two disjoint convex closed sets C, D there exist convex closed sets C′, D′, with

scholarly journals Convex hull property and maximum principle for finite element minimisers of general convex functionals

2013 ◽  
Vol 124 (4) ◽  
pp. 685-700 ◽  
Author(s):  
Lars Diening ◽  
Christian Kreuzer ◽  
Sebastian Schwarzacher
Keyword(s):  
Finite Element ◽  
Maximum Principle ◽  
Convex Hull ◽  
Convex Functionals ◽  
General Convex

Support functions of general convex sets

2003 ◽  
Vol 49 (3) ◽  
pp. 305-319
Author(s):  
Dug-Hwan Choi ◽  
Jonathan D. H. Smith
Keyword(s):  
Convex Sets ◽  
Support Functions ◽  
General Convex

The Power of Weighted Regularity-Preserving Multi Bottom-Up Tree Transducers

2015 ◽  
Vol 26 (07) ◽  
pp. 987-1005 ◽  
Author(s):  
Andreas Maletti
Keyword(s):  
Machine Translation ◽  
Statistical Machine Translation ◽  
Expressive Power ◽  
Proof Technique ◽  
Top Down ◽  
Bottom Up ◽  
Tree Transducer ◽  
Negative Side ◽  
Look Ahead ◽  
Tree Transducers

The expressive power of regularity-preserving [Formula: see text]-free weighted linear multi bottom-up tree transducers is investigated. These models have very attractive theoretical and algorithmic properties, but (especially in the weighted setting) their expressive power is not well understood. Despite the regularity-preserving restriction, their power still exceeds that of composition chains of [Formula: see text]-free weighted linear extended top-down tree transducers with regular look-ahead. The latter devices are a natural super-class of weighted synchronous tree substitution grammars, which are commonly used in syntax-based statistical machine translation. In particular, the linguistically motivated discontinuous transformation of topicalization can be modeled by such multi bottom-up tree transducers, whereas the mentioned composition chains cannot implement it. On the negative side, the inverse of topicalization cannot be implemented by any such multi bottom-up tree transducer, which confirms their bottom-up nature (and non-closure under inverses). An interesting, promising, and widely applicable proof technique is used to prove these statements.

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