Convex Analysis and Nonlinear Optimization

Abstract

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 publi cation of trade names, trademarks , service mark s, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whethe r or not they are subject to proprietary right s.Printed in the United States of America. 987 6 5 4 3 2 I springeronline.com (MVY) To our families PrefaceOptimization is a rich a nd t hriving mathem ati cal di scipline. P roperties of minimizers and m aximiz ers of functions rely inti m ately on a wealth of t echniques fro m m athem a t ical analysis, including tools from ca lculus and its generalizat ions, t opological no tions, and mor e geometric id eas. T he theory underl ying curren t computational op timization t echniques grows ever more sophisticated -duali ty-based algor ithms, interior point methods, and cont rol-theoretic app licati ons are typi cal examp les. The powerful and elegant langu age of convex analysis unifi es mu ch of t his t heory. Hen ce our aim of writing a concis e, accessible account of convex analys is a nd its applicat ions and exte ns ions , for a broad audience .For stude nts of op timization and an alysis , there is gre at b en efit t o blurrin g the distinction between t he two disciplines. Many important analytic problems have illuminating optimization formulations and hence can b e approached through our main variat ion al tools: subgradients and optimality condit ions, t he many guises of duality, metric regul arity and so for th. More generally, t he ide a of convexi ty is central t o the transition from classical analysis to various branches of modern analysis: fro m lin ear to nonlinear analysis, from smoot h to nonsmooth, and from the study of functions to mul tifunction s. Thus, although we use cert ain optimization models repe atedly to illustrate the main results (models such as linear and semidefinit e programming du ali ty and cone pol arity) , we constant ly emphasize the power of abstract models and notation.Good refer ence works on finit e-dimensional convex a nalysis already exist . Rockafellar's classic Conv ex Analysis [167] has b een indispensable and ubiquitous since t he 1970s, and a more general sequel with Wets, Variational Analysis [168], app eared recently. Hiriart-Urruty and Lemar ech al 's Convex Analysis and Minimization Algorithms [97] is a compreh ensive but gentler introduction. Our goal is not t o suppla nt these works, but on the cont rary to promote them , a nd ther eby to motivate future res ear chers. This book aims to make converts. vii viii Preface We t ry to be succinct rather t han systematic, avo id ing b ecoming bo gged down in tec hnical det ails. Our style is relatively infor mal ; for exam ple, t he text of each sect ion creates the context for many of t he resul t stateme n...

Fifty years of maximal monotonicity

“…If one considers a linear program min c, x s.t. Ax ≤ b, x ≥ 0 with respect to coordinate orderings then the associated variational equality [9,Sect. 7.3] relies on the skew matrix…”

Parametric Borwein-Preiss variational principle and applications

“…( Note that part (ii) (c) of the above corollary in finite-dimensional spaces is a better result than those obtained by applying Cellina's approximate selection theorem [7], Theorem 8.2.5, to the subdifferential of f . Cellina's theorem gives a continuous mapping c : M → E such that c(y) ∈ ∂f (y + δB) + δB.…”

“…(i) The assertion follows from (v) of Theorem 4.2, having in mind that ∂dist C(p) (x) = B ∩ N C(p) (x) (see, for instance, [7], page 57, Exercise 12 (d) (iii)).…”

Douglas-Rachford feasibility methods for matrix completion problems

“…4 The complete structure of these proteins is contained in the respective data files as a list of points in R 3 , each representing an individual atom. The corresponding complete Euclidean distance matrix can then be computed using (14). A realistic partial Euclidean distance matrix is then obtained by removing all entries which correspond to distances greater than 6Å.…”