On Simplicial Longest Edge Bisection in Lipschitz Global Optimization

Herrera, Juan F. R.; Casado, Leocadio G.; Hendrix, E.M.T.; Garcı́a, Inmaculada

doi:10.1007/978-3-319-09129-7_8

Cited by 3 publications

(3 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The LEB α longest edge selection is not unique for a 4-simplex, thus further investigation is needed. The behavior of LEB α longest edge selection applied over an existing B&B algorithm has been tested in (Herrera et al, 2014). The results show that although the LEB α selection rule is computationally more complex than a rule just selecting the first longest edge, the wall clock time of the algorithm is reduced considerably due to the reduction in the number of evaluated simplices.…”

Section: Resultsmentioning

confidence: 99%

On the Minimum Number of Simplex Shapes in Longest Edge Bisection Refinement of a Regular <italic>n</italic>-Simplex

Aparicio¹,

Casado²,

Hendrix³

et al. 2015

Informatica

Self Cite

View full text Add to dashboard Cite

Abstract. In several areas like Global Optimization using branch-and-bound methods, the unit n-simplex is refined by bisecting the longest edge such that a binary search tree appears. This process generates simplices belonging to different shape classes. Having less simplex shapes facilitates the prediction of the further workload from a node in the binary tree, because the same shape leads to the same sub-tree. Irregular sub-simplices generated in the refinement process may have more than one longest edge when n 3. The question is how to choose the longest edge to be bisected such that the number of shape classes is as small as possible. We develop a Branch-and-Bound (B&B) algorithm to find the minimum number of classes in the refinement process. The developed B&B algorithm provides a minimum number of eight classes for a regular 3-simplex. Due to the high computational cost of solving this combinatorial problem, future research focuses on using high performance computing to derive the minimum number of shapes in higher dimensions.

show abstract

Section: Resultsmentioning

confidence: 99%

On the Minimum Number of Simplex Shapes in Longest Edge Bisection Refinement of a Regular <italic>n</italic>-Simplex

Aparicio¹,

Casado²,

Hendrix³

et al. 2015

Informatica

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this context, our initial question was about the effect when one applies different heuristics to simplicial B&B on a box constrained area, where the initial search region is not a regular simplex and the termination criterion is based on the bounding rule. In a previous study [7], we showed that the number of evaluated simplices can be reduced by not selecting the first longest edge, but that which has the smallest sum of angles with the other edges. That study generated the upper part of a binary tree running a Lipschitz B&B with a rough accuracy.…”

Section: Introductionmentioning

confidence: 96%

Heuristics for Longest Edge Selection in Simplicial Branch and Bound

Herrera

Casado

Hendrix

et al. 2015

Computational Science and Its Applications -- ICCSA 2015

Self Cite

View full text Add to dashboard Cite

Simplicial partitions are suitable to divide a bounded area in branch and bound. In the iterative refinement process, a popular strategy is to divide simplices by their longest edge, thus avoiding needle-shaped simplices. A range of possibilities arises in higher dimensions where the number of longest edges in a simplex is greater than one. The behaviour of the search and the resulting binary search tree depend on the selected longest edge. In this work, we investigate different rules to select a longest edge and study the resulting efficiency of the branch and bound algorithm.

show abstract

“…One of the questions is how to determine the sequence of LEs to be bisected which generates the smallest number of simplices in the complete BT. Several heuristics have been presented (Aparicio et al 2014) and have been applied to Global Optimization algorithms in Herrera et al (2014). Determination of the set of indices of bisected LEs (one per simplex) to generate one of the smallest BT requires a complete enumeration of all possible LEB combinations.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2015

View full text Add to dashboard Cite

In several areas like global optimization using branch-and-bound methods for mixture design, the unit n-simplex is refined by longest edge bisection (LEB). This process provides a binary search tree. For n > 2, simplices appearing during the refinement process can have more than one longest edge (LE). The size of the resulting binary tree depends on the specific sequence of bisected longest edges. The questions are how to calculate the size of one of the smallest binary trees generated by LEB and how to find the corresponding sequence of LEs to bisect, which can be represented by a set of LE indices. Algorithms answering these questions are presented here. We focus on sets of LE indices that are repeated at a level of the binary tree. A set of

show abstract

On Simplicial Longest Edge Bisection in Lipschitz Global Optimization

Cited by 3 publications

References 12 publications

On the Minimum Number of Simplex Shapes in Longest Edge Bisection Refinement of a Regular <italic>n</italic>-Simplex

On the Minimum Number of Simplex Shapes in Longest Edge Bisection Refinement of a Regular <italic>n</italic>-Simplex

Heuristics for Longest Edge Selection in Simplicial Branch and Bound

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

Contact Info

Product

Resources

About