Generating a smallest binary tree by proper selection of the longest edges to bisect in a unit simplex refinement

Salmeron, Jose L.; Aparicio, Guillermo; Casado, Leocadio G.; Garcı́a, Inmaculada; Hendrix, E.M.T.; G.-Tóth, Boglárka

doi:10.1007/s10878-015-9970-y

Cited by 8 publications

(4 citation statements)

References 9 publications

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“…When other lower bound methods are added to IA, the evaluation of simplex vertices can be disabled in order to save computation. However, this may imply another (worse) update off and a different course of the algorithm, due to Longest Edge Bisection (LEB) by the first longest edge, instead of the best LEB [19].…”

Section: Remarkmentioning

confidence: 99%

On new methods to construct lower bounds in simplicial branch and bound based on interval arithmetic

et al. 2021

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Branch and Bound (B&B) algorithms in Global Optimization are used to perform an exhaustive search over the feasible area. One choice is to use simplicial partition sets. Obtaining sharp and cheap bounds of the objective function over a simplex is very important in the construction of efficient Global Optimization B&B algorithms. Although enclosing a simplex in a box implies an overestimation, boxes are more natural when dealing with individual coordinate bounds, and bounding ranges with Interval Arithmetic (IA) is computationally cheap. This paper introduces several linear relaxations using gradient information and Affine Arithmetic and experimentally studies their efficiency compared to traditional lower bounds obtained by natural and centered IA forms and their adaption to simplices. A Global Optimization B&B algorithm with monotonicity test over a simplex is used to compare their efficiency over a set of low dimensional test problems with instances that either have a box constrained search region or where the feasible set is a simplex. Numerical results show that it is possible to obtain tight lower bounds over simplicial subsets.

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Section: Remarkmentioning

confidence: 99%

On new methods to construct lower bounds in simplicial branch and bound based on interval arithmetic

et al. 2021

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“…This section describes an algorithm for determining the size of the smallest binary tree generated using Longest Edge Bisection as the rule for subdividing simplices as published in Salmerón et al (2015). The method applies a full enumeration of simplices checking every division option, i.e.…”

Section: Minimum Tree Sizementioning

confidence: 99%

“…First of all, heuristics were developed to choose the longest edge and the size of the tree was measured in Aparicio et al (2014) taking as a reference the "first" longest edge as choice rule. Next, a specific algorithm was developed to discover the minimum size of a tree in dimension n + 1 given an accuracy ǫ in Salmerón et al (2015). Following the line of early research of Adler (1983), Horst (1997), Hendrix et al (2012), we studied bisection of the longest edge in Euclidean norm.…”

Section: Introductionmentioning

confidence: 99%

On Bisecting the Unit Simplex Using Various Distance Norms

Salmeron¹,

Casado²,

Hendrix³

2016

Informatica

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The iterative bisection of the longest edge of the unit simplex generates a binary tree, where the specific shape depends on the chosen longest edges to be bisected. In global optimization, the use of various distance norms may be advantageous for bounding purposes. The question dealt with in this paper is how the size of a binary tree generated by the refinement process depends on heuristics for longest edge selection when various distance norms are used. We focus on the minimum size of the tree that can be reached, how selection criteria may reduce the size of the tree compared to selecting the first edge, whether a predefined grid is covered and how unique are the selection criteria. The exact numerical values are provided for the unit simplex in 4 and 5-dimensional space.

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“…Recently, the regular tetrahedron has received much attention in areas such as global optimization for mixture design by using the so-called branch-and-bound methods, where the regular n-simplex is bisected [19,20]. Additionally, this tetrahedron can be found in the triangulation of the 3D cube into five tetrahedra, and the building of the nearly equilateral tetrahedra carried out by Adler in 1983 [21] was also based on the regular tetrahedron.…”

Section: Introductionmentioning

confidence: 99%

Similarity Classes in the Eight-Tetrahedron Longest-Edge Partition of a Regular Tetrahedron

Padrón,

Plaza,

Suárez

2023

Mathematics

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A tetrahedron is called regular if its six edges are of equal length. It is clear that, for an initial regular tetrahedron R0, the iterative eight-tetrahedron longest-edge partition (8T-LE) of R0 produces an infinity sequence of tetrahedral meshes, τ0={R0}, τ1={Ri1}, τ2={Ri2},…, τn={Rin},…. In this paper, it is proven that, in the iterative process just mentioned, only two distinct similarity classes are generated. Therefore, the stability and the non-degeneracy of the generated meshes, as well as the minimum and maximum angle condition straightforwardly follow. Additionally, for a standard-shape tetrahedron quality measure (η) and any tetrahedron Rin∈τn, n>0, then ηRin≥23ηR0. The non-degeneracy constant is c=23 in the case of the iterative 8T-LE partition of a regular tetrahedron.

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Generating a smallest binary tree by proper selection of the longest edges to bisect in a unit simplex refinement

Cited by 8 publications

References 9 publications

On new methods to construct lower bounds in simplicial branch and bound based on interval arithmetic

On new methods to construct lower bounds in simplicial branch and bound based on interval arithmetic

On Bisecting the Unit Simplex Using Various Distance Norms

Similarity Classes in the Eight-Tetrahedron Longest-Edge Partition of a Regular Tetrahedron

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