Global Optimization Deterministic Approaches
The main contents and character of the monograph did not change with respect to the first edition. However, within most chapters we incorporated quite a number of modifications which take into account the recent development of the field, the very valuable
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Springer-Verlag Berlin Heidelberg GmbH
Reiner Horst· Hoang Tuy
Global Optimization Deterministic Approaches Third Revised and Enlarged Edition
With 55 Figures and 7 Tables
Springer
Professor Dr. Remer Horst University ofTrier Department of Mathematics P.O.Box 3825 D-54286 Trier, Germany Professor Dr. Hoang Tuy Vien Toan Hoc Institute ofMathematics P.O.Box 631, BO-HO 10000 Hanoi, Vietnam
Cataloging-in-Publication Data applied for Die Deutsche Bibllothek - CIP-Einheitsaufnahme
Hor.t, Relner: Global optlmization : determinlstic approaches / Reiner Horst ; Hoang Tuy. - 3., rev. and enl. ed. ISBN 978-3-642-08247-4 ISBN 978-3-662-03199-5 (eBook) DOI 10.1007/978-3-662-03199-5 NE: Tuy, Hoang:
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerne 0 Vx e D. Let i e D. Define for keIN r(i, k):=
[!(!l] 1> f(i) r
k
dx.
Then, it is shown in Falk (1973a) that i is a global maximizer of f over D if and only if the sequence {r(i, k)}kelN is bounded. Although it is not very practical for solving the general global optimization problem, similar ideas have been used by Zhirov (1985) und Duong (1987) to
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propose algorithms for globally minimizing a polynomial over a parallelepiped, respeetively, a polytope. Global optimality eonditions for special problems ean be found, e.g., in Hiriart-Urruty (1989) (differenees of eonvex funetions), and in Warga (1992) (quadratic problems). Note that eertain important classes of optimization problems have the property that every loeal minimum is a global minimum. A well-known example is eonvex minimization where the objeetive funetion is a eonvex funetion and where the feasible set is a eonvex set. Sinee, in these classes, standard optimization proeedures for finding loeal solutions will yield the global optimum, eonsiderable effort has gone into charaeterizing families of funetions and feasible sets having the property that every loeal minimum is a global solution. For a number of results on this question see Martos (1967), Mangasarian (1969), Martos (1975), Zang and Avriel (1975), Netzer and Passy (1975), Zang et al. (1976), Avriel (1976), Avriel and Zang (1981), Horst (1982 and 1984a,b), Gasanov and Rikun (1984 and 1985), Horst and Thaeh (1988). Throughout this volume global optimization problems are eonsidered where
standard optimization techniques Jau because oJthe ezistence oJlocal minima that are not global. These global optimization problems will be ealled multiextremal global optimization problems. Due to the inherent diffieulties mentioned above, the methods devised for analysing multiextremal global optimization problems are quite diverse and signifieantly different from the standard tools referred to above. Though several general theoretieal eoneepts exist for solving problem (1), in order to build a numerieally promising implementation, additional properties of the problems data usually have to be exploited. Convexity, for example, will often be present. Many problems have linear eonstraints. Other problems i
