On global optimization using interval arithmetic
type: Item
instance of: scholarly article
title: On global optimization using interval arithmetic
zbMATH Open document ID: 0774.65036
DOI: 10.1007/BF02238641
author: H. Munack
published in: Computing
publication date: 1993-01-11T00:00:00Z
review text: Consider an objective function \(f:\mathbb{R}^ n \to\mathbb{R}\) and a box \(B\) in \(\mathbb{R}^ n\). The author aims at the global minimizers of \(f\) in \(B\). An interval analysis based algorithm for the stated problem has been described by \textit{E. R. Hansen} [J. Optimization Theory Appl. 29, 331-344 (1979; Zbl 0388.65023) and Numer. Math. 34, 247-270 (1980; Zbl 0442.65052)]. Hansen's method requires an interval extension \(F\) of the objective function \(f\). The author investigates the case where such an interval extension \(F\) is not available or too expensive. He approximates the objective function \(f\) by a model function \(m\) which piecewise has the form \(m(x)=\text{sinh}(x^ T Ax+b^ T x+c)\) with \(A\in\mathbb{R}^{n\times n}\), \(b\in\mathbb{R}^ n\), \(c\in\mathbb{R}\). Then he applies Hansen's method to the model function \(m\). The quantities \(A\), \(b\), \(c\) are determined using only function values of \(f\). The main disadvantage of this approach is that the global minimizers of \(m\) are only approximations of the global minimizers of \(f\). Nevertheless, numerical tests with objective functions from magnetic optics yield results of practical relevance.
reviewed by: Herbert Fischer
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 65G30
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 86184
zbMATH Keywords: global optimization
zbMATH Keywords: interval arithmetic
zbMATH Keywords: inclusion function
zbMATH Keywords: interval analysis
zbMATH Keywords: numerical tests
describes a project that uses: ALGOL 60
describes a project that uses: TOMS659
MaRDI profile type: Publication
cites work: An economic method of computing LPτ-sequences
cites work: The algorithmic language Triplex-ALGOL 60
cites work: Algorithm 659
cites work: Global optimization using interval analysis - the multi-dimensional case
cites work: Global optimization using interval analysis: The one-dimensional case
cites work: Q4732095
cites work: Q5566070
cites work: On computing the range of a rational function of n variables over a bounded region
cites work: On irregularities of distribution
cites work: On the distribution of points in a cube and the approximate evaluation of integrals
cites work: Uniformly distributed sequences with an additional uniform property
cites work: On the Systematic Search in a Hypercube
full work available at URL: https://doi.org/10.1007/bf02238641
OpenAlex ID: W190239871
description: scientific article; zbMATH DE number 86184