Foundations of Optimization
The book gives a detailed and rigorous treatment of the theory of optimization (unconstrained optimization, nonlinear programming, semi-infinite programming, etc.) in finite-dimensional spaces. The fundamental results of convexity theory and the theory of
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Osman Güler
Foundations of Optimization
Osman Güler Department of Mathematics & Statistics University of Maryland, Baltimore County Hilltop Circle 1000 Baltimore, MD 21250 USA [email protected]
Editorial Board: S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
K. A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720 USA [email protected]
ISSN 0072-5285 ISBN 978-0-387-34431-7 e-ISBN 978-0-387-68407-9 DOI 10.1007/978-0-387-68407-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010932783 Mathematics Subject Classification (2010): 90-01, 90C30, 90C46, 90C25, 90C05, 90C20, 90C34, 90C47, 49M37, 49N15, 49J53, 49J50, 49M15, 49K35, 65K05, 65K10, 52A05, 52A07, 52A35, 52A41, 52A40, 52A37 © Springer Science +Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
I dedicate this book to my mother, Z¨ ulfiye G¨ uler, and to the memory of my father, E¸sref G¨ uler.
Preface
Optimization is everywhere. It is human nature to seek the best option among all that are available. Nature, too, seems to be guided by optimization—many laws of nature have a variational character. Among geometric figures in the plane with a fixed perimeter, the circle has the greatest area. Such isoperimetric problems involving geometric figures date back to ancient Greece. Fermat’s principle, discovered in 1629, stating that the tangent line is horizontal at a minimum point, seems to have influenced the development of calculus. The proofs of Rolle’s theorem and the mean value theorem in calculus use the Weierstrass theorem on the existence of maximizers and minimizers. The introduction of the brachistochrone problem in 1696 by Johann Bernoulli had a tremendous impact on the development of the calculus of variations and influenced the development of functional analysis. The variational character of laws of mechanics and optics were discovered in the seventeenth and eighteenth centuries. Euler and Lagrange forged the foundations of the calculus of variations in the eighteenth century. In the nineteenth century, Riemann used Dirichlet’s principle, which has a varia
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