Variational Analysis
From its origins in the minimization of integral functionals, the notion of 'variations' has evolved greatly in connection with applications in optimization, equilibrium, and control. It refers not only to constrained movement away from a point, but also
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Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics
Series editors M. Berger P. de la Harpe F. Hirzebruch N.J. Hitchin L. Hörmander A. Kupiainen G. Lebeau M. Ratner D. Serre Y.G. Sinai N.J.A. Sloane A. M. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates
S.R.S. Varadhan
317
R. Tyrrell Rockafellar • Roger J-B Wets
Variational Analysis with figures drawn by Maria Wets
ABC
R. Tyrrell Rockafellar Department of Mathematics University of Washington Seattle, WA 98195-4350 USA [email protected]
Roger J-B Wets Department of Mathematics University of California at Davis One Shields Ave. Davis, CA 95616 USA [email protected]
ISSN 0072 -7830 ISBN 978-3-540-62772-2 e-ISBN 978-3-642-02431-3 DOI 10.1007/978-3-642-02431-3 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009929711 Mathematics Subject Classification (2000): 47H05, 49J40, 49J45, 49J52, 49K40, 49N15, 52A50, 52A41, 54B20, 54C60, 54C65, 90C31 c Springer-Verlag Berlin Heidelberg 1998, Corrected 3rd printing 2009
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PREFACE In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. For a long time, ‘variational’ problems have been identified mostly with the ‘calculus of variations’. In that venerable subject, built around the minimization of integral functionals, constraints were relatively simple and much of the focus was on infinite-dimensional function spaces. A major theme was the exploration of variations around a point, within the bounds imposed by the constraints, in order to help characterize solutions and portray them in terms of ‘variational principles’. Notions of perturbation, approximation and even generalized differentiability were extensively investigated. Variational theory progressed also to the study of so-called stationary
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