mathematical developments arising from linear … · 47 linear algebra and its role in ... 69...

Mathematical Developments Arising from Linear Programming

American Mathematical Society

Titles in This Series

Volume 1 Markov random fields and their

applications, Ross Kindermann and J. Laurie Snell

Proceedings of the Northwestern homotopy theory conference, Haynes R. Miller and Stewart B. Priddy, Editors

2 Proceedings of the conference on integration, topology, and geometry in linear spaces, William H. Graves, Editor

Low dimensional topology, Samuel J. Lomonaco, Jr., Editor

Topological methods in nonlinear functional analysis, S. P. Singh, S. Thomeier, and B. Watson, Editors

3 The closed graph and P-closed graph properties in general topology, T. R. Hamlett and L. L. Herrington Factorizations of bn & 1, b = 2,

3,5,6,7,10,11,12 up to high powers, John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff, Jr.

4 Problems of elastic stability and vibrations, Vadim Komkov, Editor

5 Rational constructions of modules for simple Lie algebras, George B. Seligman

6 Umbral calculus and Hopf algebras, Robert Morris, Editor

Chapter 9 of Ramanujan's second notebook-Infinite series identities, transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi 7 Complex contour integral

representation of cardinal spline functions, Walter Schempp

Central extensions, Galois groups, and ideal class groups of number fields, A. Frohlich 8 Ordered fields and real algebraic

geometry, D. W. Dubois and T. Recio, Editors

Value distribution theory and its applications, Chung-Chun Yang, Editor

Conference in modern analysis and probability, Richard Beals, Anatole Beck, Alexandra Bellow, and Arshag Hajian, Editors

9 Papers in algebra, analysis and statistics, R. Lidl, Editor

10 Operator algebres and K-theory, Ronald G. Douglas and Claude Schochet, Editors Microlocal analysis, M. Salah Baouendi,

Richard Beals, and Linda Preiss Rothschild, Editors

11 Plane ellipticity and related problems, Robert P. Gilbert, Editor

Fluids and plasmas: geometry and dynamics, Jerrold E. Marsden, Editor

12 Symposium on algebraic topology in honor of Jose Adem, Samuel Gitler, Editor Automated theorem proving, W. W.

Bledsoe and Donald Loveland, Editors 13 Algebraists' homage: Papers in ring theory and related topics, S. A. Amitsur, D. J. Saltman, and G. B. Seligman, Editors

14 Lectures on Nielsen fixed point theory, Boju Jiang

Mathematical applications of category theory, J. W. Gray, Editor

Axiomatic set theory, James E. Baumgartner, Donald A. Martin, and Saharon Shelah, Editors

15 Advanced analytic number theory. Part I: Ramification theoretic methods, Carlos J. Moreno

Proceedings of the conference on Banach algebras and several complex variables, F. Greenleaf and D. Gulick, Editors 16 Complex representations of GL(2, K) for

finite fields K, llya Piatetski-Shapiro Contributions to group theory, Kenneth I. Appel, John G. Ratcliffe, and Paul E. Schupp, Editors

17 Nonlinear partial differential equations, Joel A. Smoller, Editor

18 Fixed points and nonexpansive mappings, Robert C. Sine, Editor

Combinatorics and algebra, Curtis Greene, Editor

http://dx.doi.org/10.1090/conm/114


Volume 35 Four-manifold theory, Cameron Gordon

and Robion Kirby, Editors

36 Group actions on manifolds, Reinhard Schultz, Editor

37 Conference on algebraic topology in honor of Peter Hilton, Renzo Piccinini and Denis Sjerve, Editors

38 Topics in complex analysis, Dorothy Browne Shaffer, Editor

39 Errett Bishop: Reflections on him and his research, Murray Rosenblatt, Editor

40 Integral bases for affine Lie algebras and their universal enveloping algebras, David Mitzman

41 Particle systems, random media and large deviations, Richard Durrett, Editor

42 Classical real analysis, Daniel Waterman, Editor

43 Group actions on rings, Susan Montgomery, Editor

44 Combinatorial methods in topology and algebraic geometry, John R. Harper and Richard Mandelbaum, Editors

45 Finite groups-coming of age, John McKay, Editor

46 Structure of the standard modules for the affine Lie algebra A?), James Lepowsky and Mirko Primc

47 Linear algebra and its role in systems theory, Richard A. Brualdi, David H. Carlson, Biswa Nath Datta, Charles R. Johnson, and Robert J. Plemmons, Editors

48 Analytic functions of one complex variable, Chung-chun Yang and Chi-tai Chuang, Editors

49 Complex differential geometry and nonlinear differential equations, Yum-Tong Siu, Editor

50 Random matrices and their applications, Joel E. Cohen, Harry Kesten, and Charles M. Newman, Editors

51 Nonlinear problems in geometry, Dennis M. DeTurck, Editor

52 Geometry of normed linear spaces, R. G. Bartle, N. T. Peck, A. L. Peressini, and J. J. Uhl, Editors

53 The Selberg trace formula and related topics, Dennis A. Hejhal, Peter Sarnak, and Audrey Anne Terras, Editors

54 Differential analysis and infinite dimensional spaces, Kondagunta Sundaresan and Srinivasa Swaminathan, Editors

55 Applications of algebraic K-theory to algebraic geometry and number theory, Spencer J. Bloch, R. Keith Dennis, Eric M. Friedlander, and Michael R. Stein, Editors

56 MuFparameter bifurcation theory, Martin Golubitsky and John Guckenheimer, Editors

57 Combinatorics and ordered sets, Ivan Rival, Editor

58.1 The Lefschetz centennial conference. Proceedings on algebraic geometry, D. Sundararaman, Editor

58.11 The Lefschetz centennial conference. Proceedings on algebraic topology, S. Gitler, Editor

58.111 The Lefschetz centennial conference. Proceedings on differential equations, A. Verjovsky, Editor

59 Function estimates, J. S. Marron, Editor

60 Nonstrictly hyperbolic conservation laws, Barbara Lee Keyfitz and Herbert C. Kranzer, Editors

61 Residues and traces of differential forms via Hochschild homology, Joseph Lipman

62 Operator algebras and mathematical physics, Palle E. T. Jorgensen and Paul S. Muhly, Editors

63 Integral geometry, Robert L. Bryant, Victor Guillemin, Sigurdur Helgason, and R. 0. Wells, Jr., Editors

64 The legacy of Sonya Kovalevskaya, Linda Keen, Editor

65 Logic and combinatorics, Stephen G. Simpson, Editor

66 Free group rings, Narian Gupta

67 Current trends in arithmetical algebraic geometry, Kenneth A. Ribet, Editor


Volume 68 Differential geometry: The interface

between pure and applied mathematics, Mladen Luksic, Clyde Martin, and William Shadwick, Editors

69 Methods and applications of mathematical logic, Walter A. Carnielli and Luiz Paulo de Alcantara, Editors

70 Index theory of elliptic operators, foliations, and operator algebras, Jerome Kaminker, Kenneth C. Millett, and Claude Schochet, Editors

71 Mathematics and general relativity, James A. Isenberg, Editor

72 Fixed point theory and its applications, R. F. Brown, Editor

73 Geometry of random motion, Rick Durrett and Mark A. Pinsky, Editors

74 Geometry of group representations, William M. Goldman and Andy R. Magid, Editors

75 The finite calculus associated with Bessel functions, Frank M. Cholewinski

76 The structure of finite algebras, David C. Hobby and Ralph Mckenzie

77 Number theory and its applications in China, Wang Yuan, Yang Chung-chun, and Pan Chengbiao, Editors

78 Braids, Joan S. Birman and Anatoly Libgober, Editors

79 Regular differential forms, Ernst Kunz and Rolf Waldi

80 Statistical inference from stochastic processes, N. U. Prabhu, Editor

81 Hamiltonian dynamical systems, Kenneth R. Meyer and Donald G. Saari, Editors

82 Classical groups and related topics, Alexander J. Hahn, Donald G. James, and Zhe-xian Wan, Editors

83 Algebraic K-theory and algebraic number theory, Michael R. Stein and R. Keith Dennis, Editors

84 Partition problems in topology, Stevo Todorcevic

85 Banach space theory, Bor-Luh Lin, Editor

86 Representation theory and number theory in connection with the local Langlands conjecture, J. Ritter, Editor

87 Abelian group theory, Laszlo Fuchs, Rudiger Gobel, and Phillip Schultz, Editors

88 Invariant theory, R. Fossum, W. Haboush, M. Hochster, and V. Lakshmibai, Editors

89 Graphs and algorithms, R. Bruce Richter, Editor

90 Singularities, Richard Randell, Editor

91 Commutative harmonic analysis, David Colella, Editor

92 Categories in computer science and logic, John W. Gray and Andre Scedrov, Editors

93 Representation theory, group rings, and coding theory, M. Isaacs, A. Lichtman, D. Passman, S. Sehgal, N. J. A. Sloane, and H. Zassenhaus, Editors

94 Measure and measurable dynamics, R. Daniel Mauldin, R. M. Shortt, and Cesar E. Silva, Editors

95 Infinite algebraic extensions of finite fields, Joel V. Brawley and George E. Schnibben

96 Algebraic topology, Mark Mahowald and Stewart Priddy, Editors

97 Dynamics and control of multibody systems, J. E. Marsden, P. S. Krishnaprasad, and J. C. Simo, Editors

98 Every planar map is four colorable, Kenneth Appel and Wolfgang Haken

99 The connection between infinite dimensional and finite dimensional dynamical systems, Basil Nicolaenko, Ciprian Foias, and Roger Temam, Editors

100 Current progress in hyperbolic systems: Riemann problems and computations, W. Brent Lindquist, Editor

101 Recent developments in geometry, S.-Y. Cheng, H. Choi, and Robert E. Greene, Editors

102 Primes associated to an ideal, Stephen McAdam

103 Coloring theories, Steve Fisk


Volume 104 Accessible categories: The foundations

of categorical model theory, Michael Makkai and Robert Pare

105 Geometric and topological invariants of elliptic operators, Jerome Kaminker, Editor

106 Logic and computation, Wilfried Sieg, Editor

107 Harmonic analysis and partial differential equations, Mario Milman and Tomas Schonbek, Editors

108 Mathematics of nonlinear science, Melvyn S. Berger, Editor

109 Combinatorial group theory, Benjamin Fine, Anthony Gaglione, and Francis C. Y. Tang, Editors

110 Lie algebras and related topics, Georgia Benkart and J. Marshall Osborn, Editors

11 1 Finite geometries and combinatorial designs, Earl S. Kramer and Spyros S. Magliveras, Editors

112 Statistical analysis of measurement error models and applications, Philip J. Brown and Wayne A. Fuller, Editors

113 Integral geometry and tomography, Eric Grinberg and Eric Todd Quinto, Editors

114 Mathematical developments arising from linear programming, Jeffrey C. Lagarias and Michael J. Todd, Editors

ATHEMATICS


Proceedings of a Joint Summer Research Conference held at Bowdoin College, June 25-July 1, 1988

Jeffrey C . Lagarias and Michael J. Todd, Editors

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND

EDITORIAL BOARD

Richard W. Beals, managing editor Sylvain E. Cappell Jan Mycielski Jonathan Goodman Linda Preiss Rothschild Craig Huneke Michael E. Taylor

The AMS-IMS-SIAM Joint Summer Research Conference on Mathematical Developments Arising from Linear Programming was held at Bowdoin College, Brunswick, Maine, on June 25-July 1, 1988, with support from the National Sci- ence Foundation, Grant DMS-8746804, the U.S. Army Research Office, Grant DMS- 8845058, and the Office of Naval Research, Grant 11 11 /87/A0544.

This work relates to Department of Navy, Grant N00014-88-J-1019 issued by the Office of Naval Research. The United States Government has a royalty-free license throughout the world in all copyrightable material contained herein.

1980 Mathematics Subject Classification (1985 Revision). Primary 90C, 65K, 49D; Sec- ondary 52A25,58F07.

Library of Congress Cataloging-in-Publication Data

AMS-IMS-SIAM Joint Summer Research Conference on Mathematical Developments Arising from Linear Programming (1988: Bowdoin College)

Mathematical developments arising from linear programming: proceedings of the AMS-IMS-SIAM joint summer research conference held June 25-July 1, 1988, with support from the National Science Foundation, the U.S. Army Research Office, and the Office of Naval ResearchiJeffrey C. Lagarias and Michael J. Todd, editors.

p. cm.-(Contemporary mathematics, ISSN 0271 -41 32; 11 4) Includes bibliographical references. ISBN 0-821 8-51 21-7 1. Mathematical programming-Congresses. 2. Linear programming-Congresses.

I. Lagarias, Jeffrey C., 1949- . II. Todd, Michael J., 1947- . Ill. Title. IV. Series: Contemporary mathematics (American Mathematical Society); v. 114. QA402.5.A454 1988 90-22942 51 9.7'2-dc20 CIP

Copyright @ 1990, American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted

to the United States Government. Printed in the United States of America.

Information on copying and reprinting can be found at the back of this volume. The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability. @ This publication was typeset using AMS-TEX,

the American Mathematical Society's T$ macro system.

Contents

1. Recent Progress and New Directions

Some Recent Results on Convex Polytopes CARL W. LEE

Probabilistic Analysis of the Simplex Method KARL HEINZ BORGWARDT

On Solving the Linear Programming Problem Approximately NIMROD MEGIDDO

Riemannian Geometry Underlying Interior-Point Methods for Linear Programming

NARENDRA KARMARKAR

Steepest Descent, Linear Programming, and Hamiltonian Flows A. M. BLOCH

2. Interior-Point Methods for Linear Programming

An O(n3L) Potential Reduction Algorithm for Linear Programming YINYU YE

I. I. Dikin's Convergence Result for the Affine-Scaling Algorithm R. J. VANDERBEI AND J. C. LAGARIAS

Phase 1 Search Directions for a Primal-Dual Interior Point Method for Linear Programming

IRVIN J. LUSTIG

Some Results Concerning Convergence of the Affine Scaling Algorithm EARL R. BARNES

Dual Ellipsoids and Degeneracy in the Projective Algorithm for Linear Programming

KURT M. ANSTREICHER

x CONTENTS

A Note on Limiting Behavior of the Projective and the Affine Rescaling Algorithms

MIROSLAV D. ASIC, VERA V. KOVA~EVIC-VUJ~IC, AND MIRJANA D. RADOSAVLJEVIC-NIKOLIC 15 1

3. Trajectories of Interior-Point Methods On the Convergence Behavior of Trajectories for Linear Programming

CHRISTOPH WITZGALL, PAUL T. BOGGS, AND PAUL D. DOMICH 161

Limiting Behavior of the Affine Scaling Continuous Trajectories for Linear Programming Problems

ILAN ADLER AND RENATO D. C. MONTEIRO 189

Convergence and Boundary Behavior of the Projective Scaling Trajec- tories for Linear Programming

RENATO D. C. MONTEIRO 2 13

4. Nonlinear Optimization On the Complexity of a Numerical Algorithm for Solving Generalized

Convex Quadratic Programs by Following a Central Path F. JARRE, G. SONNEVEND, AND J. STOER 233

Canonical Problems for Quadratic Programming and Projective Meth- ods for Their Solution

BAHMAN KALANTARI 243

An Interior Point Algorithm for Solving Smooth Convex Programs Based on Newton's Method

SANJAY MEHROTRA AND JIE SUN 265

A Modified Kantorovich Inequality for the Convergence of Newton's Method

A. A. GOLDSTEIN 285

5. Integer Programming and Multi-Objective Programming An Interior-Point Approach to NP-complete Problems-Part I

NARENDRA KARMARKAR 297

Solving Matching Problems Using Karmarkar's Algorithm JOHN E. MITCHELL AND MICHAEL J. TODD

Efficient Faces of Polytopes: Interior Point Algorithms, Parameter- ization of Algebraic Varieties, and Multiple Objective Optimization

S. S. ABHYANKAR, T. L. MORIN, AND T. TRAFALIS 3 19

Preface

This volume contains the proceedings of the AMS Summer Research Con- ference on Mathematical Developments Arising from Linear Programming held at Bowdoin College, June 25-July 1, 1988. This conference presented current research in linear and nonlinear programming and related areas of mathematics. There has been intense work in this area, much of it in extend- ing and understanding the ideas underlying N. Karmarkar's interior-point linear programming algorithm, which was proposed in 1984. This research effort is interdisciplinary, and the conference brought together mathemati- cians, computer scientists, and operations researchers.

The state of the field in 1987 is illustrated in Progress in Mathematical Pro- gramming, Interior-Point and Related Methods (N. Megiddo, ed.), Springer- Verlag, Berlin and New York, 1989. To put the results presented in this volume in perspective, we first review the state of knowledge at that time.

Karmarkar's algorithm is an interior-point method for solving linear programs. It requires as input a linear program provided with a special initial starting point in the interior of the polytope of feasible solutions, called the center. There are two related types of algorithms, the projective scaling algorithm, which uses projective transformations, that Karmarkar proved to be a polynomial time algorithm, and the afine scaling algorithm, which uses affine transformations, and which has not, in general, been proved to converge in polynomial time (and probably does not). The affine scaling algorithm, how- ever, has computational advantages in practice, and many of the computer implementations of "Karmarkar's algorithm" actually use affine scaling ideas. For both algorithms there is a vector field on the polytope of feasible solutions, which yields differential equations giving trajectories of feasible solutions inside the polytope all of which go to an optimal solution. The affine and projective scaling methods have different trajectories in general, but they have one trajectory in common, the central trajectory or central path, which turns out also to be a logarithmic barrier function trajectory previously studied in connection with nonlinear programming algorithms. Karmarkar's method,

xii PREFACE

and many subsequent algorithms, approximately follow the central trajectory. Karmarkar originally proved that a linear program (in equality form) in n dimensions and with input size L takes O(nL) iterations to converge and o ( ~ ~ . ~ L ) arithmetic operations in total. J. Renegar, using path-following ideas, found an algorithm requiring at most O ( f i L ) iterations. By early 1987 P. Vaidya and, independently, C. Gonzaga had obtained methods that followed the central trajectory requiring 0 ( n 3 ~ ) arithmetic operations in total. Karmarkar's algorithm also used nonlinear programming ideas involving minimizing a "potential function." Such ideas carry over to give a polynomial time algorithm for convex quadratic programming as was shown by S. Kapoor and P. Vaidya in 1987, and for a class of linear complementarity problems by M. Kojima, S. Mizuno, and A. Yoshise. Finally, much work was done on developing computationally efficient versions of interior-point methods, including variants of the projective scaling algorithm that can han- dle problems in standard form and which do not require advance knowledge of the optimal objective function value.

Now we describe the results presented in this volume. The conference had ten invited talks which were intended to provide broad views of recent work in various areas related to linear programming. Section 1 presents papers based on such talks. The remaining sections present contributed papers classified by subject area, most of which use interior-point ideas.

The papers in Section 1 reflect the wide range of areas of mathematics on which linear programming impinges. Convex polytopes are the basic mathematical objects underlying linear programming problems and the simplex method. The paper of C. L. Lee describes recent results on the combinatorial structure of convex polytopes. K. H. Borgwardt surveys "average-case" polynomial running time bounds obtained for variants of the simplex method under various probability models. The paper of N. Megiddo analyzes several different notions of "approximate solution" of a linear programming problem. N. Karmarkar presents new results that use Riemannian geometric methods to study the behavior of trajectories underlying interior-point methods. His results yield insights concerning the running time of such algorithms. Fi- nally the paper of A. Bloch surveys gradient-like flows arising from several algorithms, including interior-point linear programming algorithms, the QR- algorithm for diagonalizing symmetric matrices, and methods for the total least squares problem, and matching problems. It shows that these flows are related to completely integrable Hamiltonian systems by suitable nonlinear transformations.

Section 2 presents results on interior-point methods for linear programming. The paper of Y. Ye gives an O(Ji iL) iteration algorithm using a new class of potential functions that apparently does not require staying close to the central trajectory. This is a significant advance, because the previously known O(+L) algorithms take "small" steps, so that such algorithms must take on the order of f i L iterations to get close to an optimal solution. The

PREFACE xiii

idea of Ye allows algorithms that can greedily take bigger steps and still the worst-case analysis applies. (In practical implementations one takes bigger steps than the complexity analyses allow.) Ye has since shown that those ideas extend to "projective" algorithms. In another direction, in early 1988 the linear programming community in the West discovered that the affine scaling algorithm was proposed in 1967 by the Soviet mathematician I. I. Dikin, and that he published a proof of convergence for it in 1974. R. J. Vanderbei and J. C. Lagarias give an expose of Dikin's proof of convergence, which applies under the assumption of primal nondegeneracy. One of the difficulties of interior-point methods is that they require linear programs to be transformed to a form that comes with an initial interior feasible solution. The paper of I. Lustig presents a new method for obtaining an initial feasible point for a linear program and its dual. E. Barnes discusses another method for obtaining a feasible starting point for the affine scaling algorithm and gives a convergence result for this algorithm. The paper of K. Anstre- icher considers ellipsoids containing dual optimal solutions for a projective scaling interior point method in the case of primal degeneracy. M. ASiC, V. KovaEeviC-VujEic, and M. RadosavljeviC-Nikoli analyze the asymptotic behavior of Karmarkar's algorithm, obtaining results valid for degenerate linear programs. They also propose a rounding method to go from an interior feasible point to the exact optimal solution.

Section 3 presents results on the trajetories determined by "infinitesimal" versions of the affine scaling and projective scaling algorithms. The paper of C. Witzgall, P. Boggs, and P. Domich and the paper of I. Adler and R. Mon- teiro both prove that limiting behavior exists for affine scaling trajectories. Their results apply even to degenerate linear programs, including cases where the convergence of the affine scaling algorithm has not yet been proved. The paper of R. C. Monteiro analyzes the boundary behavior of projective scaling trajectories.

Section 4 presents results for nonlinear programming problems. F. Jarre, G. Sonnevend, and J. Stoer give a polynomial-time interior point method for solving convex quadratic programming problems having convex quadratic constraints. B. Kalantari studies the problem of finding a zero of a quadratic form over a simplex, a problem which is NP-complete in general. He gives a "projective" algorithm for finding a local minimum of a certain potential function, which gives a polynomial-time algorithm for solving certain convex quadratic programs including linear programming. S. Mehrotra and J. Sun present an interior-point method for smooth convex programming. A. Gold- stein gives a criterion specifying a quadratic convergence region when using Newton's method to find a zero of a nonlinear function.

Section 5 presents results for integer programming and multi-objective programming. The paper of N. Karmarkar gives an interior-point approach to solving 0-1 integer programming problems. Such problems, which are NP- complete in general, are converted to nonconvex quadratic programs on a

xiv PREFACE

hypercube. Karmarkar isolates a subclass of such problems for which the set of optimal solutions is connected. He suggests that this approach will solve a large class of 0-1 integer programs not previously considered tractable, including many set covering problems. J. Mitchell and M. Todd study per- fect matching problems, which are a class of integer programs known to be solvable in polynomial time. They describe a cutting plane method for such problems that uses interior-point methods to solve linear programming re- laxations of the problem and present computational data. Finally, the paper of S. Abhyankar, T. Morin, and T. Trafalis outlines two methods for solving multi-objective linear programs, including an interior-point method to find a single efficient solution, and a method of circumscribed algebraic sets to find the entire set of efficient solutions.

We would like to take this opportunity to thank the anonymous referees. We also thank the National Science Foundation and Office of Naval Research for their support of the conference. The breadth of topics covered owed much to the valuable advice of the organizing committee, consisting of Victor Klee and Steve Smale. Finally, the success of the meeting owed much to the excellent local arrangements and support of the AMS staff, in particular Ms. Carole Kohanski.

The papers in this volume are in final form and no version will be submit- ted for publication elsewhere, except for the paper by I ~ i n J. Lustig and the paper by John E. Mitchell and Michael J. Todd.

Jeffrey C. Lagarias Michael J. Todd

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mathematical developments arising from linear … · 47 linear algebra and its role in ... 69...

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