Convex Optimization: Stephen Boyd and Lieven Vandenberghe

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Convex Optimization

Convex Optimization

Stephen Boyd
Department of Electrical Engineering
Stanford University
Lieven Vandenberghe
Electrical Engineering Department
University of California, Los Angeles

cambridge university press
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York http://www.cambridge.org
Information on this title: www.cambridge.org/9780521833783
c Cambridge University Press 2004
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2004
Seventh printing with corrections 2009
Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloguing-in-Publication data
Boyd, Stephen P.
Convex Optimization / Stephen Boyd & Lieven Vandenberghe
p. cm.
Includes bibliographical references and index.
ISBN 0 521 83378 7
1. Mathematical optimization. 2. Convex functions. I. Vandenberghe, Lieven. II. Title. QA402.5.B69 2004
519.6–dc22
2003063284
ISBN 978-0-521-83378-3 hardback

Cambridge University Press has no responsiblity for the persistency or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

For
Anna, Nicholas, and Nora
Dani¨l and Margriet
e

Contents
Preface
1 Introduction
1.1 Mathematical optimization . . . . . .
1.2 Least-squares and linear programming
1.3 Convex optimization . . . . . . . . . .
1.4 Nonlinear optimization . . . . . . . .
1.5 Outline . . . . . . . . . . . . . . . . .
1.6 Notation . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . .

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1
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Theory

2 Convex sets
2.1 Affine and convex sets . . . . . . . . .
2.2 Some important examples . . . . . . .
2.3 Operations that preserve convexity . .
2.4 Generalized inequalities . . . . . . . .
2.5 Separating and supporting hyperplanes
2.6 Dual cones and generalized inequalities
Bibliography . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . .

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