Quasidifferentiability and Related Topics
2 Radiant sets 236 3 Co-radiant sets 239 4 Radiative and co-radiative sets 241 5 Radiant sets with Lipschitz continuous Minkowski gauges 245 6 Star-shaped sets and their kernels 249 7 Separation 251 8 Abstract convex star-shaped sets 255 References 260 11
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Nonconvex Optimization and Its Applications Volume 43 Managing Editor: Panos Pardalos University of Florida, U.S.A.
Advisory Board: Ding-Zhu Du University of Minnesota, U.S.A. C. A. Floudas Princeton University, U.S.A.
J. Mockus Stanford University, U.S.A. H. D. Sherali Virginia Polytechnic Institute and State University, U.S.A.
The titles published in this series are listed at the end of this volume.
Quasidifferentiability and Related Topics Edited by
Vladimir Demyanov Department of Applied Mathematics, St. Petersburg State University, St. Petersburg, Russia
and
Alexander Rubinov School of Information Technology and Mathematical Sciences, University of Ballarat, Victoria, Australia
Springer-Science+Business Media, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-4830-4 ISBN 978-1-4757-3137-8 (eBook) DOl 10.1007/978-1-4757-3137-8
Printed on acid-free paper
All Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000. Softcover reprint of the hardcover 1st edition 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
to Professor Franco Giannessi on his 65th birthday and to Professor Diethard Pallaschke on his 60th birthday
vii
Quasidifferentiability and Related Topics Many nondifferentiable functions, which arise in various applications, possess the property of quasidifferentiability. The class of quasidiferentiable functions represents a linear space closed with respect to all algebraic operations, compositions and the operations of taking the pointwise maximum and minimum. Each quasidifferentiable function is characterized by its quasidifferential - a pair of compact sets in the dual space. Quasidifferentiable functions enjoy a rich calculus of quasidifferentials which is a generalization of classical (smooth) Calculus. The present book is a collection of, mostly, surveys of some recent results on different aspects of Nonsmooth Analysis related to, connected with or inspired by Quasidifferential Calculus (QD Calculus). Some applications to various problems of Mechanics and Mathematics are discussed, numerical algorithms are described and compared. Open problems still remained are exposed and stated. The book is aimed at providing up-to-date information concerning quasi differentiable functions and related topics. The State-of-the-Art in Quasidifferential Calculus is viewed and evaluated by experts, both researchers and users. Quasidifferentiable functions were introduced in 1979 and the twentieth anniversary of this development provides a good occasion to appraise the impact, results and perspectives of the field.
Contents
Preface
xv
References
xix
1
1
AN INTRODUCTION TO QUASI DIFFERENTIAL CALCULUS V.F.Demyanov, A.M.Rubinov 1 Introduct
