# Subdivision, Sampling, and Initialization Strategies for Simplical Branch and Bound in Global Optimization

Jens Clausen, A, Zilinskas

Research output: Contribution to journalJournal articleResearchpeer-review

### Abstract

We consider the problem of optimizing a Lipshitzian function. The branch and bound technique is a well-known solution method, and the key components for this are the subdivision scheme, the bound calculation scheme, and the initialization. For Lipschitzian optimization, the bound calculations are based on the sampling of function values.

We propose a branch and bound algorithm based on regular simplexes. Initially, the domain in question is covered with regular simplexes, and our subdivision scheme maintains this property. The bound calculation becomes both simple and efficient, and we describe two schemes for sampling points of the function: midpoint sampling and vertex sampling.

The convergence of the algorithm is proved, and numerical results are presented for the two dimensional case, for which also a special initial covering is presented. (C) 2002 Elsevier Science Ltd. All rights reserved.
Original language English Computers & Mathematics with Applications 44 7 943-955 0898-1221 https://doi.org/10.1016/S0898-1221(02)00205-5 Published - 2002

### Cite this

@article{fc8f155d99874cf5953f6e1fc3db8e8f,
title = "Subdivision, Sampling, and Initialization Strategies for Simplical Branch and Bound in Global Optimization",
abstract = "We consider the problem of optimizing a Lipshitzian function. The branch and bound technique is a well-known solution method, and the key components for this are the subdivision scheme, the bound calculation scheme, and the initialization. For Lipschitzian optimization, the bound calculations are based on the sampling of function values.We propose a branch and bound algorithm based on regular simplexes. Initially, the domain in question is covered with regular simplexes, and our subdivision scheme maintains this property. The bound calculation becomes both simple and efficient, and we describe two schemes for sampling points of the function: midpoint sampling and vertex sampling.The convergence of the algorithm is proved, and numerical results are presented for the two dimensional case, for which also a special initial covering is presented. (C) 2002 Elsevier Science Ltd. All rights reserved.",
keywords = "Branch and Bound, Global Optimization",
author = "Jens Clausen and A, Zilinskas",
year = "2002",
doi = "10.1016/S0898-1221(02)00205-5",
language = "English",
volume = "44",
pages = "943--955",
journal = "Computers & Mathematics with Applications",
issn = "0898-1221",
publisher = "Pergamon Press",
number = "7",

}

Subdivision, Sampling, and Initialization Strategies for Simplical Branch and Bound in Global Optimization. / Clausen, Jens; Zilinskas, A,.

In: Computers & Mathematics with Applications, Vol. 44, No. 7, 2002, p. 943-955.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Subdivision, Sampling, and Initialization Strategies for Simplical Branch and Bound in Global Optimization

AU - Clausen, Jens

AU - Zilinskas, A,

PY - 2002

Y1 - 2002

N2 - We consider the problem of optimizing a Lipshitzian function. The branch and bound technique is a well-known solution method, and the key components for this are the subdivision scheme, the bound calculation scheme, and the initialization. For Lipschitzian optimization, the bound calculations are based on the sampling of function values.We propose a branch and bound algorithm based on regular simplexes. Initially, the domain in question is covered with regular simplexes, and our subdivision scheme maintains this property. The bound calculation becomes both simple and efficient, and we describe two schemes for sampling points of the function: midpoint sampling and vertex sampling.The convergence of the algorithm is proved, and numerical results are presented for the two dimensional case, for which also a special initial covering is presented. (C) 2002 Elsevier Science Ltd. All rights reserved.

AB - We consider the problem of optimizing a Lipshitzian function. The branch and bound technique is a well-known solution method, and the key components for this are the subdivision scheme, the bound calculation scheme, and the initialization. For Lipschitzian optimization, the bound calculations are based on the sampling of function values.We propose a branch and bound algorithm based on regular simplexes. Initially, the domain in question is covered with regular simplexes, and our subdivision scheme maintains this property. The bound calculation becomes both simple and efficient, and we describe two schemes for sampling points of the function: midpoint sampling and vertex sampling.The convergence of the algorithm is proved, and numerical results are presented for the two dimensional case, for which also a special initial covering is presented. (C) 2002 Elsevier Science Ltd. All rights reserved.

KW - Branch and Bound

KW - Global Optimization

U2 - 10.1016/S0898-1221(02)00205-5

DO - 10.1016/S0898-1221(02)00205-5

M3 - Journal article

VL - 44

SP - 943

EP - 955

JO - Computers & Mathematics with Applications

JF - Computers & Mathematics with Applications

SN - 0898-1221

IS - 7

ER -