Lecture Notes in Economics and Mathematical Systems 637 Founding Editors: M Beckmann H.P Künzi Managing Editors: Prof Dr G Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr 140/AVZ II, 58084 Hagen, Germany Prof Dr W Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitätsstr 25, 33615 Bielefeld, Germany Editorial Board: H Dawid, D Dimitrow, A Gerber, C-J Haake, C Hofmann, T Pfeiffer, R Slowiński, W.H.M Zijm For further volumes: http://www.springer.com/series/300 Radu Ioan Bot¸ Conjugate Duality in Convex Optimization 123 Dr.rer.nat.habil Radu Ioan Bot¸ Chemnitz University of Technology Faculty of Mathematics Reichenhainer Str 39 09126 Chemnitz Germany radu.bot@mathematik.tu-chemnitz.de ISSN 0075-8442 ISBN 978-3-642-04899-9 e-ISBN 978-3-642-04900-2 DOI 0.1007/978-3-642-04900-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009943057 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permissions for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To my family Preface The results presented in this book originate from the last decade research work of the author in the field of duality theory in convex optimization The reputation of duality in the optimization theory comes mainly from the major role that it plays in formulating necessary and sufficient optimality conditions and, consequently, in generating different algorithmic approaches for solving mathematical programming problems The investigations made in this work prove the importance of the duality theory beyond these aspects and emphasize its strong connections with different topics in convex analysis, nonlinear analysis, functional analysis and in the theory of monotone operators The first part of the book brings to the attention of the reader the perturbation approach as a fundamental tool for developing the so-called conjugate duality theory The classical Lagrange and Fenchel duality approaches are particular instances of this general concept More than that, the generalized interior point regularity conditions stated in the past for the two mentioned situations turn out to be particularizations of the ones given in this general setting In our investigations, the perturbation approach represents the starting point for deriving new duality concepts for several classes of convex optimization problems Moreover, via this approach, generalized Moreau–Rockafellar formulae are provided and, in connection with them, a new class of regularity conditions, called closedness-type conditions, for both stable strong duality and strong duality is introduced By stable strong duality we understand the situation in which strong duality still holds whenever perturbing the objective function of the primal problem with a linear continuous functional The closedness-type conditions constitute a class of regularity conditions recently introduced in the literature They experience at present an increasing interest in the optimization community, as they are widely applicable than the generalized interior point ones, a fact that we also point out in this work We employ the conjugate duality in establishing biconjugate formulae for different classes of convex functions and, in the special case of Fenchel duality, we offer some deep insights into the existing relations between the notions strong and stable strong duality Moreover, we enlarge the class of generalized interior point regularity conditions given for both Fenchel and Lagrange duality approaches by formulating corresponding sufficient conditions expressed via the quasi-interior and quasi-relative interior vii viii Preface The convex analysis and, especially, the duality theory have surprisingly found in the last years applications in rediscovering classical results and also in giving new powerful ones in the field of monotone operators Among others, we provide a regularity condition of closedness-type for the maximality of the sum of two maximal monotone operators in reflexive Banach spaces, which proves to be weaker than all the other generalized interior point conditions introduced in the literature with the same purpose I express my thanks to Gert Wanka for his incessant support and for giving me the possibility to my research in such a nice academic environment like Chemnitz is offering Thanks also to Ioana Costantea, Ernăo Robert Csetnek, SorinMihai Grad, Andre Heinrich, Ioan Bogdan Hodrea, Altangerel Lkhamsuren, Nicole Lorenz, Andreea Moldovan and Emese Tăunde Vargyas, former and current members of the research group at the university in Chemnitz, with whom I share not only an intense scientific collaboration but also a nice friendship I am indebted to Ernăo Robert Csetnek, Anca Grad and Sorin-Mihai Grad for reading preliminary versions of this work and for their suggestions and remarks I am grateful to my parents and to my sister whose encouragement and support I felt all the time even if they are far away Finally, I want to thank my wife Nina and my daughter Cassandra Maria for their love, patience and understanding Chemnitz, Germany, August, 2009 Radu Ioan Bot¸ Contents Introduction I Perturbation Functions and Dual Problems A General Approach for Duality The Problem Having the Composition with a Linear Continuous Operator in the Objective Function 14 The Problem with Geometric and Cone Constraints 19 The Composed Convex Optimization Problem 28 II Moreau–Rockafellar Formulae and Closedness-Type Regularity Conditions Generalized Moreau–Rockafellar Formulae Stable Strong Duality for the Composed Convex Optimization Problem Stable Strong Duality for the Problem Having the Composition with a Linear Continuous Operator in the Objective Function Stable Strong Duality for the Problem with Geometric and Cone Constraints Closedness Regarding a Set III IV Biconjugate Functions 10 The Biconjugate of a General Perturbation Function 11 Biconjugates Formulae for Different Classes of Convex Functions 12 The Supremum of an (Infinite) Family of Convex Functions 13 The Supremum of Two Convex Functions Strong and Total Conjugate Duality 14 A General Closedness–Type Regularity Condition for (Only) Strong Duality 15 Strong Fenchel Duality 16 Strong Lagrange and Fenchel–Lagrange Duality 17 Total Lagrange and Fenchel–Lagrange Duality 35 35 38 44 50 56 65 65 68 73 82 87 87 89 93 99 ix x Contents V Unconventional Fenchel Duality 105 18 Totally Fenchel Unstable Functions .105 19 Totally Fenchel Unstable Functions in Finite Dimensional Spaces .112 20 Quasi Interior and Quasi-relative Interior .115 21 Regularity Conditions via qi and qri .119 22 Lagrange Duality via Fenchel Duality .127 VI Applications of the Duality to Monotone Operators .133 23 Monotone Operators and Their Representative Functions 133 24 Maximal Monotonicity of the Operator S C A TA .136 25 The Maximality of A TA and S C T .142 26 Enlargements of Monotone Operators .148 References .157 Index 163 Symbols and Notations Sets, Elements and Relations X !.X; X / !.X ; X / b x BX int.U / cl.U / lin.U / aff.U / co.U / cone.U / coneco.U / core.U / icr.U / sqri.U / ri.U / qri.U / qi.U / X m NU" x/ NU x/ TU x/ ÄC 1C Z C RT RT Topological dual space of X Weak topology on X induced by X Weak topology on X induced by X The canonical image in X of the element x X The unit ball of the normed space X Interior of the set U Closure of the set U Linear hull of the set U Affine hull of the set U Convex hull of the set U Conic hull of the set U Convex conic hull of the set U Algebraic interior of the set U Intrinsic core of the set U Strong quasi-relative interior of the set U Relative interior of the set U Quasi-relative interior of the set U Quasi interior of the set U The set f.x; : : : ; x/ X m W x X g "-normal set to the set U at x Normal cone to the set U at x Bouligand tangent cone to the set U at x The partial ordering induced by the convex cone C A greatest element with respect to the ordering cone C attached to a space The space Z to which the element 1C is added Dual cone of the cone C The space of all functions z W T ! R The set of the constant functions z RT xi ... dealing with semi-infinite programming problems (see, for instance, [64, 74, 75]) In the lines of the investigations made by Bot¸, Grad and Wanka in [30, 31], in Section 17 we continue treating... of the author in the field of duality theory in convex optimization The reputation of duality in the optimization theory comes mainly from the major role that it plays in formulating necessary... in the optimization community, as they are widely applicable than the generalized interior point ones, a fact that we also point out in this work We employ the conjugate duality in establishing