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Interior Point Methods of Mathematical Programming Applied Optimization Interior Point Methods of Mathematical Programming Applied Optimization Volume 5 Series Editors: Panos M. Pardalos University of Florida, U.SA. Donald Hearn University ofFlorida, U.S.A. The titles published in this series are listed at the end of this volume. Interior Point Methods of Mathematical Programming Edited by Tamas Terlaky Delft University a/Technology KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-13: 978-1-4613-3451-4 e-ISBN-13: 978-1-4613-3449-1 DOl: 10.1007/978-1-4613-3449-1 Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus NiJ'hoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved @ 19% Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. This book is dedicated to the memory of Professor Gyorgy Sonnevend, the father of analytic centers. CONTENTS PREFACE xv Part I LINEAR PROGRAMMING 1 1 INTRODUCTION TO THE THEORY OF INTERIOR POINT METHODS Benjamin Jansen, Cornelis Roos, Tamas Terlaky 3 1.1 The Theory of Linear Programming 3 1.2 Sensitivity Analysis in Linear Programming 14 1.3 Concluding Remarks 30 REFERENCES 31 2 AFFINE SCALING ALGORITHM Takashi Tsuchiya 35 2.1 Introduction 35 2.2 Problem and Preliminaries 38 2.3 The Affine Scaling Algorithm 40 2.4 Nondegeneracy Assumptions 47 2.5 Basic Properties of the Iterative Process 50 2.6 Global Convergence Proof Under a Nondegeneracy Assumption 54 2.7 Global Convergence Proof Without Nondegeneracy Assumptions 56 2.8 The Homogeneous Affine Scaling Algorithm 59 2.9 More on the Global Convergence Proof of the Affine Scaling Algorithm 67 2.10 Why Two-Thirds is Sharp for the Affine Scaling? 68 2.11 Superlinear Convergence of the Affine Scaling Algorithm 69 2.12 On the Counterexample of Global Convergence of The Affine Scaling Algorithm 70 VB Vlll INTERIOR POINT METHODS IN MATHEMATICAL PROGRAMMING 2.13 Concluding Remarks 73 2.14 Appendix: How to Solve General LP Problems with the Affine Scaling Algorithm 75 REFERENCES 77 3 TARGET-FOLLOWING METHODS FOR LINEAR PROGRAMMING Benjamin Jansen, Cornelis Roos, Tamas Terlaky 83 3.1 Introduction 83 3.2 Short-step Primal-dual Algorithms for LP 86 3.3 Applications 93 3.4 Concluding Remarks 121 REFERENCES 121 4 POTENTIAL REDUCTION ALGORITHMS Kurt M. Anstreicher 125 4.1 Introduction 125 4.2 Potential Functions for Linear Programming 126 4.3 Karmarkar's Algorithm 130 4.4 The Affine Potential Reduction Algorithm 134 4.5 The Primal-Dual Algorithm 139 4.6 Enhancements and Extensions 142 REFERENCES 151 5 INFEASIBLE-INTERIOR-POINT ALGORITHMS Shinji Mizuno 159 5.1 Introduction 159 5.2 An lIP Algorithm Using a Path of Centers 161 5.3 Global Convergence 164 5.4 Polynomial Time Convergence 172 5.5 An lIP Algorithm Using a Surface of Centers 175 5.6 A Predictor-corrector Algorithm 178 5.7 Convergence Properties 181 5.8 Concluding Remarks 184 REFERENCES 185 Contents IX 6 IMPLEMENTATION OF INTERIOR-POINT METHODS FOR LARGE SCALE LINEAR PROGRAMS Erling D. Andersen, Jacek Gondzio, Csaba Meszaros, Xiaojie Xu 189 6.1 Introduction 190 6.2 The Primal-dual Algorithm 193 6.3 Self-dual Embedding 200 6.4 Solving the Newton Equations 204 6.5 Presolve 225 6.6 Higher Order Extensions 230 6.7 Optimal Basis Identification 235 6.8 Interior Point Software 240 6.9 Is All the Work Already Done? 243 6.10 Conclusions 244 REFERENCES 245 Part II CONVEX PROGRAMMING 253 7 INTERIOR-POINT METHODS FOR CLASSES OF CONVEX PROGRAMS Florian Jarre 255 7.1 The Problem and a Simple Method 256 7.2 Self-Concordance 258 7.3 A Basic Algorithm 281 7.4 Some Applications 291 REFERENCES 293 8 COMPLEMENTARITY PROBLEMS Akiko Yoshise 297 8.1 Introduction 297 8.2 Monotone Linear Complementarity Problems 300 8.3 Newton's Method and the Path of Centers 308 8.4 Two Prototype Algorithms for the Monotone LCP 316 8.5 Computational Complexity of the Algorithms 332 x INTERIOR POINT METHODS IN MATHEMATICAL PROGRAMMING 8.6 Further Developments and Extensions 339 8.7 Proofs of Lemmas and Theorems 345 REFERENCES 359 9 SEMIDEFINITE PROGRAMMING Motakuri V. Ramana, Panos M. Pardalos 369 9.1 Introduction 369 9.2 Geometry and Duality 370 9.3 Algorithms and Complexity 377 9.4 Applications 383 9.5 Concluding Remarks 390 REFERENCES 391 10 IMPLEMENTING BARRIER METHODS FOR NONLINEAR PROGRAMMING David F. Shanno, Mark G. Breitfeld, Evangelia M. Simantiraki 399 10.1 Introduction 399 10.2 Modified Penalty-Barrier Methods 402 10.3 A Slack Variable Alternative 407 10.4 Discussion and Preliminary Numerical Results 411 REFERENCES 413 Part III APPLICATIONS, EXTENSIONS 415 11 INTERIOR POINT METHODS FOR COMBINATORIAL OPTIMIZATION John E. Mitchell 417 11.1 Introduction 417 11.2 Interior Point Branch and Cut Algorithms 419 11.3 A Potential Function Method 441 11.4 Solving Network Flow Problems 445 11.5 The Multicommodity Network Flow Problem 451 11.6 Computational Complexity Results 455 11.7 Conclusions 457 REFERENCES 459 Contents Xl 12 INTERIOR POINT METHODS FOR GLOBAL OPTIMIZATION Panos M. Pardalos, Mauricio G. C. Resende 467 12.1 Introduction 467 12.2 Quadratic Programming 468 12.3 Nonconvex Potential Function Minimization 474 12.4 Affine Scaling Algorithm for General Quadratic Programming 486 12.5 A Lower Bounding Technique 490 12.6 Nonconvex Complementarity Problems 493 12.7 Concluding Remarks 497 REFERENCES 497 13 INTERIOR POINT APPROACHES FOR THE VLSI PLACEMENT PROBLEM Anthony Vannelli, Andrew Kennings, Paulina Chin 501 13.1 Introduction 501 13.2 A Linear Program Formulation of the Placement Problem 503 13.3 A Quadratic Program Formulation of the MNP Placement Model 509 13.4 Towards Overlap Removal 512 13.5 Primal-Dual Quadratic Interior Point Methods 514 13.6 Numerical Results 518 13.7 Conclusions 524 REFERENCES 526 CONTRIBUTORS Erling D. Andersen Benjamin Jansen Department of Management Faculty of Technical Mathematics Odense University and Computer Science Campusvej 55 Delft University of Technology DK-5230 Odense M, Denmark Mekelweg 4, 2628 CD, Delft e-mail: [email protected] The Netherlands e-mail: [email protected] Kurt M. Anstreicher School of Business Administration Florian J arre The University of Iowa Institut fiir Angewandte Iowa City, Iowa 52242, USA Mathematik und Statistik e-mail: [email protected]. uiowa.edu U niversitat W iirz burg 97074 Wiirzburg, Germany Mark G. Breitfeld e-mail: [email protected] A.T. Kearny, GmbH. Stuttgart, Germany Andrew Kennings Department of Electrical and Paulina Chin Computer Engineering Department of Electrical and University of Waterloo Computer Engineering Waterloo, Ontario, CANADA N2L 3G1 University of Waterloo e-mail: [email protected] Waterloo, Ontario, CANADA N2L 3G1 e-mail: [email protected] Csaba Meszaros Department of Operations Research J acek Gondzio and Decision Support Systems Logilab, HEC Geneva Computer and Automation Institute Section of Management Studies Hungarian Academy of Sciences University of Geneva Lagymanyosi u. 11 102 Bd Carl Vogt Budapest, Hungary CH-1211 Geneva 4, Switzerland [email protected] e-mail: [email protected] (on leave from the John E. Mitchell Systems Research Institute Department of Mathematical Sciences Polish Academy of Sciences Rensselaer Polytechnic Institute Newelska 6, 01-447 Troy, NY 12180, USA Warsaw, Poland) e-mail: [email protected] xiii XIV CONTRIBUTORS Shinji Mizuno Evangelia M. Simantiraki Department of Prediction and Control RUTCOR and The Institute of Statistical Mathematics Graduate School of Management Minata-ku, Tokyo 106, Japan Rutgers University e-mail: [email protected] New Brunswick, New Jersey, USA Panos M. Pardalos e-mail: [email protected] Department of Industrial and Systems Engi­ neering Tamas Terlaky 303 Well Hall Faculty of Technical Mathematics University of Florida and Computer Science Gainesville Delft University of Technology Florida, FL 32611-9083 USA Mekelweg 4, 2628 CD, Delft e-mail: [email protected] The Netherlands e-mail: [email protected] Motakuri V. Ramana Department of Industrial and Systems Engi­ Takashi Tsuchiya neering The Institute of Statistical Mathematics 303 Wei! Hall Department of Prediction and Control University of Florida 4-6-7 Minami-Azabu, Minata-ku, Gainesville Tokyo 106, Japan Florida, FL 32611-9083 USA e-mail: [email protected] e-mail: [email protected] Anthony Vannelli Mauricio G.C. Resende Department of Electrical and AT&T Bell Laboratories Computer Engineering Murray Hill University of Waterloo New Jersey 09794 USA Waterloo, Ontario, CANADA N2L 3Gl e-mail: [email protected] e-mail: [email protected] Cornelis Roos Xiaojie Xu Faculty of Technical Mathematics X_Soft and Computer Science P.O. Box 7207 Delft University of Technology University, MS 38677-7207, USA Mekelweg 4, 2628 CD, Delft (on leave from the The Netherlands Institute of Systems Science e-mail: [email protected] Chinese Academy of Sciences David F. Shanno Beijing 100080, China) e-mail:[email protected] RUTCOR, Rutgers University New Brunswick, Akiko Yoshise New Jersey, USA Institute of Socia-Economic Planning e-mail: [email protected] University of Tsukuba Tsukuba, Ibaraki 305, Japan e-mail: [email protected] PREFACE One has to make everything as simple as possible but, never more simple.
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