Non-Linear Parametric Optimization
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B. Bank, J. Guddat, D. Klatte,
B. Kummer and K. Tammer
Non-Linear Parametric Optimization
1983
Springer Fachmedien Wiesbaden GmbH
Library of Congress Cataloging in Publication Data Main entry under title: Non-linear parametric optimization. Bibliography: p. Includes index. I. Mathematical optimization. r. Bank. B. (Bernd), 19411982 511 82-17761 QA402.5.N63
CIP-Kurztitelaufnahme der Deutschen Bibliothek Non-linear parametric optimization / by B. Bank ... - Basel: Boston ; Stuttgart; Birkhauser. 1983. NE: Bank, B. [Mitverf.]
Die vorliegende Publikation ist urheberrechtlich ges 1). The mapping tp is not lower semicontinuous since, for example, the limit of the distance between the solution x = 100 for A = 0 and the solution sets tp(A) for A 0 is obviously not zero.
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Important contributions to the theory of qualitative stability of parametric optimization problems are due to C. BERGE 1959 [2], G. DEBREU 1959 [1], G. DANTZIG, J. FOLKMAN, and N. SHAPIRO 1967 [1], J. P. EVANS and F. J. GOULD 1970 [1], R. R. MEYER 1970 [1], H. J. GREENBERG and W. P. PIERSKALLA 1972 [1], 1975 [2], E. G. GOL'STEIN 1971 [3], W. KRABS 1972 [1], 1973, [2], [3], W. W. HOGAN 1973 [3], S. M. ROBINSON 1975 [4], 1976 [6], 1979 [10], S. DOLECxI 1977 [2], 1978 [3], [4], B. BROSOWSKI 1976 [1], 1980 [4], and other authors. We also refer the reader to B. KUMMER 1) cf. Section 2.2.
1. General Introduction
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1977 [1], [2], [3], 1978 [4], D. KLATTE 1977, 1979 [1], [2], [3], and B. BANK 1978 [1], 1979 [2]. Although they lie outside the scope of this volume further studies should be included in what we called the first line of development of parametric optimization. These include differentiability conditions for the extreme value function which are related to what is usually termed "differential stability" or "marginal value" in the literature. We refer the interested reader to H. D. MILLS 1956 [1], A. C. WILLIAMS 1963 [1], E. G. GOL'STEIN 1971 [3], E. S. LEVITIN 1974, 1976 [3], [6], V. F. DEM'JANOV and A. B. PEVNII 1972-74 [1], [2], [3], W. W. HOGAN 1973 [1], J. GAUVIN and J. W. TOLLE 1977 [1], and F. LEMPIO and H. MAURER 1979 [1]. Further studies are concerned with quantitative stability considerations relating to the Lipschitzian continuity or the rate of continuity of the mappings 'lJ'rand q; or the Kuhn-Tucker set mapping. By Kuhn-Tucker mapping we understand the mapping which assigns to each parameter A the set of Kuhn-Tucker solutions of the corresponding optimization problem. Besides the classic paper of A. J. HOFFMANN 1952 [1] we refer to work by D. W. WALKUP and R. J.-B. WETS 1969 [2], P. BRUNS 1972 [1], J. W. DANIEL 1973 [2], S. M. ROBINSON 1973, 1974 1975, 1976, 1977, 1979, 1980 [2], [3], [4], [5], [8], [9], [ll], W. HAGER 1976 [1], E. S. LEVITIN 1976 [5], and S. DOLECKI 1977, 1978 [2], [3], [4]. We also note a paper by P. KLEINMANN 1978 [1]. An important motivation for the study of qualitative and quantitative stability properties of parametric optimization problems comes from other areas of mathematics. Thus for instance the
