ADVANCES IN APPLIED MATHEMATICS AND GLOBAL OPTIMIZATION IN HONOR OF GILBERT STRANG Advances in Mechanics and Mathematics VOLUME 17 Series Editors David Y. Gao (Virginia Polytechnic Institute and State University) Ray W. Ogden (University of Glasgow) Advisory Board Ivar Ekeland (University of British Columbia, Vancouver) Tim Healey (Cornell University, USA) Kumbakonam Rajagopal (Texas A&M University, USA) Tudor Ratiu (École Polytechnique Fédérale, Lausanne) David J. Steigmann (University of California, Berkeley) Aims and Scope Mechanics and mathematics have been complementary partners since Newton’s time, and the history of science shows much evidence of the beneficial influence of these disciplines on each other. The discipline of mechanics, for this series, includes relevant physical and biological phenomena such as: electromagnetic, thermal, quantum effects, biomechanics, nanomechanics, multiscale modeling, dynamical systems, optimization and control, and computational methods. Driven by increasingly elaborate modern technological applications, the symbiotic relationship between mathematics and mechanics is continually growing. The increasingly large number of specialist journals has generated a complementarity gap between the partners, and this gap continues to widen. Advances in Mechanics and Mathematics is a series dedicated to the publication of the latest developments in the interaction between mechanics and mathematics and intends to bridge the gap by providing interdisciplinary publications in the form of monographs, graduate texts, edited volumes, and a special annual book consisting of invited survey articles. ADVANCES IN APPLIED MATHEMATICS AND GLOBAL OPTIMIZATION IN HONOR OF GILBERT STRANG Edited By David Y. Gao Virginia Polytechnic Institute, Blacksburg, VA Hanif D. Sherali Virginia Polytechnic Institute, Blacksburg, VA 123 Editors: David Y. G ao Hanif D. Sherali Department Mathematics Department Mathematics Virginia Polytechnic Institute Virginia Polytechnic Institute Blacksburg, VA 24061 Blacksburg, VA 24061 [email protected] [email protected] Series Editors: David Y. Gao Department of Mathematics Ray W. Ogden Virginia Polyt e chnic Institute Department of Mathematics Blacksburg, VA 24061 University of Glasgow [email protected] Glasgow, Scotland, UK [email protected] ISBN 978-0-387-75713-1 e-ISBN 978-0-387-75714-8 DOI 10.1007/978-0-387-75714-8 Library of Congress Control Number: 2009921139 Mathematics Subject Classification (2000): 90-00, 49-00, 74-00, 65, 74, 81, 92 ©Springer Science+Business Media, LLC 2009 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.com This book is dedicated to Professor Gilbert Strang on the occasion of his 70th birthday Gilbert Strang Gil Strang in his MIT office Contents Series Preface ................................................ xi Preface ....................................................... xiii Constrained Optimism ........................................ xv Gilbert Strang Biographical Summary of Gilbert Strang .....................xvii List of Publications of Gilbert Strang ......................... xix 1 Maximum Flows and Minimum Cuts in the Plane ........ 1 Gilbert Strang 2 Variational Principles and Residual Bounds for Nonpotential Equations ................................... 13 Giles Auchmuty 3 Adaptive Finite Element Solution of Variational Inequalities with Application in Contact Problems ....... 25 Viorel Bostan and Weimin Han 4 Time—Frequency Analysis of Brain Neurodynamics ...... 107 W. Art Chaovalitwongse, W. Suharitdamrong, and P.M. Pardalos 5 Nonconvex Optimization for Communication Networks ... 137 Mung Chiang ix x Contents 6 Multilevel (Hierarchical) Optimization: Complexity Issues, Optimality Conditions, Algorithms ............... 197 Altannar Chinchuluun, Panos M. Pardalos, and Hong-Xuan Huang 7 Central Path Curvature and Iteration-Complexity for Redundant Klee—Minty Cubes ............................ 223 Antoine Deza, Tam´as Terlaky, and Yuriy Zinchenko 8 Canonical Duality Theory: Connections between Nonconvex Mechanics and Global Optimization .......... 257 David Y. Gao and Hanif D. Sherali 9 Quantum Computation and Quantum Operations ......... 327 Stan Gudder 10 Ekeland Duality as a Paradigm ........................... 349 Jean-Paul Penot 11 Global Optimization in Practice: State of the Art and Perspectives .............................................. 377 J´anos D. Pint´er 12 Two-Stage Stochastic Mixed-Integer Programs: Algorithms and Insights ................................. 405 Hanif D. Sherali and Xiaomei Zhu 13 Dualistic Riemannian Manifold Structure Induced from Convex Functions ......................................... 437 Jun Zhang and Hiroshi Matsuzoe 14 NMR Quantum Computing ............................... 465 Zhigang Zhang, Goong Chen, Zijian Diao, and Philip R. Hemmer Series Preface As any human activity needs goals, mathematical research needs problems. –David Hilbert Mechanics is the paradise of mathematical sciences. –Leonardo da Vinci Mechanics and mathematics have been complementary partners since New- ton’s time, and the history of science shows much evidence of the beneficial influence of these disciplines on each other. Driven by increasingly elabo- rate modern technological applications, the symbiotic relationship between mathematics and mechanics is continually growing. However, the increasingly large number of specialist journals has generated a duality gap between the partners, and this gap is growing wider. Advances in Mechanics and Mathematics (AMMA) is intended to bridge the gap by providing multidisciplinary publications that fall into the two following complementary categories: 1. An annual book dedicated to the latest developments in mechanics and mathematics; 2. Monographs, advanced textbooks, handbooks, edited volumes, and selected conference proceedings. The AMMA annual book publishes invited and contributed comprehensive research and survey articles within the broad area of modern mechanics and applied mathematics. The discipline of mechanics, for this series, includes relevant physical and biological phenomena such as: electromagnetic, ther- mal, and quantum effects, biomechanics, nanomechanics, multiscale model- ing, dynamical systems, optimization and control, and computation methods. Especially encouraged are articles on mathematical and computational mod- els and methods based on mechanics and their interactions with other fields. All contributions will be reviewed so as to guarantee the highest possible sci- entific standards. Each chapter will reflect the most recent achievements in xi xii Series Preface the area. The coverage should be conceptual, concentrating on the method- ological thinking that will allow the nonspecialist reader to understand it. Discussion of possible future research directions in the area is welcome. Thus, the annual volumes will provide a continuous documentation of the most recent developments in these active and important interdisciplinary fields. Chapters published in this series could form bases from which possible AMMA monographs or advanced textbooks could be developed. Volumes published in the second category contain review/research contri- butions covering various aspects of the topic. Together these will provide an overview of the state-of-the-art in the respective field, extending from an in- troduction to the subject right up to the frontiers of contemporary research. Certain multidisciplinary topics, such as duality, complementarity, and sym- metry in mechanics, mathematics, and physics are of particular interest. The Advances in Mechanics and Mathematics series is directed to all sci- entists and mathematicians, including advanced students (at the doctoral and postdoctoral levels) at universities and in industry who are interested in mechanics and applied mathematics. David Y. Gao Ray W. Ogden Preface Complementarity and duality are closely related, multidisciplinary topics that pervade all natural phenomena, and form the basis for solving many under- lying nonconvex analysis and global optimization problems that arise in sci- ence and engineering. During the last forty years, much research has been devoted to the development of mathematical modeling, theory, and compu- tational methods in this area. The field has now matured in convex systems, especially in linear programming, engineering mechanics and design, mathe- matical physics, economics, optimization, and control. However, in nonconvex systems many fundamental problems still remain unsolved. In view of the importance of complementarity—duality theory and meth- ods in applied mathematics and mathematical programming, and in order to bridge the ever-increasing gap between global optimization and engineer- ing science, the First International Conference on Complementarity, Dual- ity, and Global Optimization (CDGO) was held at Virginia