Network Optimization: Continuous and Discrete Models

586 Pages·2016·7.19 MB·English

by  Bartsekas D.P.

Network Optimization:
Continuous and Discrete Models
Dimitri P. Bertsekas
Massachusetts Institute of Technology
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(cid:1)c 1998 Dimitri P. Bertsekas
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Publisher’s Cataloging-in-Publication Data
Bertsekas, Dimitri P.
Network Optimization: Continuous and Discrete Models
Includes bibliographical references and index
1. Network analysis (Planning). 2. Mathematical Optimization. I. Title.
T57.85.B44 1998 658.4’032-dc20 98-70298
ISBN 1-886529-02-7
ABOUT THE AUTHOR
Dimitri Bertsekas studied Mechanical and Electrical Engineering at
the National Technical University of Athens, Greece, and obtained his
Ph.D. in system science from the Massachusetts Institute of Technology.
He has held faculty positions at Stanford University and the Uni-
versity of Illinois. Since 1979 he has been teaching at the Massachusetts
Institute of Technology (M.I.T.), where he is currently McAfee Professor
of Engineering. He consults regularly with private industry and has held
editorial positions in several journals. His research spans several fields,
includingoptimization,control,large-scalecomputation,anddatacommu-
nication networks. He has written many research papers and he is the
author or coauthor of thirteen textbooks and research monographs.
Professor Bertsekas was awarded the INFORMS 1997 Prize for Re-
search Excellence in the Interface Between Operations Research and Com-
puter Science for his book ”Neuro-Dynamic Programming” (co-authored
with John Tsitsiklis), the 2000 Greek National Award for Operations Re-
search, and the 2001 ACC John R. Ragazzini Education Award. In 2001,
he was elected to the United States National Academy of Engineering.
iii
ATHENA SCIENTIFIC
OPTIMIZATION AND COMPUTATION SERIES
1. ConvexAnalysisandOptimization,byDimitriP.Bertsekas,with
Angelia Nedi´c and Asuman E. Ozdaglar, 2003, ISBN 1-886529-
45-0, 560 pages
2. Introduction to Probability by Dimitri P. Bertsekas and John
Tsitsiklis, 2002, ISBN 1-886529-40-X, 430 pages
3. DynamicProgrammingandOptimalControl, Vols.IandII,2nd
Edition, by Dimitri P. Bertsekas, 2001, ISBN 1-886529-08-6, 704
pages
4. Nonlinear Programming, 2nd Edition, by Dimitri P. Bertsekas,
1999, ISBN 1-886529-00-0, 800 pages
5. NetworkOptimization: ContinuousandDiscreteModelsbyDim-
itri P. Bertsekas, 1998, ISBN 1-886529-02-7, 608 pages
6. NetworkFlowsandMonotropicOptimizationbyR.TyrrellRock-
afellar, 1998, ISBN 1-886529-06-X, 634 pages
7. Introduction to Linear Optimization by Dimitris Bertsimas and
John N. Tsitsiklis, 1997, ISBN 1-886529-19-1, 608 pages
8. Parallel and Distributed Computation: Numerical Methods by
DimitriP.BertsekasandJohnN.Tsitsiklis,1997,ISBN1-886529-
01-9, 718 pages
9. Neuro-DynamicProgramming,byDimitriP.BertsekasandJohn
N. Tsitsiklis, 1996, ISBN 1-886529-10-8, 512 pages
10. Constrained Optimization and Lagrange Multiplier Methods, by
Dimitri P. Bertsekas, 1996, ISBN 1-886529-04-3, 410 pages
11. StochasticOptimalControl: TheDiscrete-TimeCasebyDimitri
P. Bertsekas and Steven E. Shreve, 1996, ISBN 1-886529-03-5,
330 pages
iv
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . p. 1
1.1. Graphs and Flows . . . . . . . . . . . . . . . . . . . . p. 3
1.1.1. Paths and Cycles . . . . . . . . . . . . . . . . . . p. 4
1.1.2. Flow and Divergence . . . . . . . . . . . . . . . . p. 6
1.1.3. Path Flows and Conformal Decomposition . . . . . . . p. 7
1.2. Network Flow Models – Examples . . . . . . . . . . . . . p. 8
1.2.1. The Minimum Cost Flow Problem . . . . . . . . . . p. 9
1.2.2. Network Flow Problems with Convex Cost . . . . . . . p. 16
1.2.3. Multicommodity Flow Problems . . . . . . . . . . . p. 17
1.2.4. Discrete Network Optimization Problems . . . . . . . p. 19
1.3. Network Flow Algorithms – An Overview . . . . . . . . . p. 20
1.3.1. Primal Cost Improvement . . . . . . . . . . . . . . p. 21
1.3.2. Dual Cost Improvement . . . . . . . . . . . . . . . p. 24
1.3.3. Auction . . . . . . . . . . . . . . . . . . . . . . p. 27
1.3.4. Good, Bad, and Polynomial Algorithms . . . . . . . . p. 35
1.4. Notes, Sources, and Exercises . . . . . . . . . . . . . . . p. 37
2. Shortest Path Problems . . . . . . . . . . . . . . . p. 51
2.1. Problem Formulation and Applications . . . . . . . . . . p. 52
2.2. A Generic Shortest Path Algorithm . . . . . . . . . . . . p. 57
2.3. Label Setting (Dijkstra) Methods . . . . . . . . . . . . . p. 65
2.3.1. Performance of Label Setting Methods . . . . . . . . . p. 68
2.3.2. The Binary Heap Method . . . . . . . . . . . . . . p. 69
2.3.3. Dial’s Algorithm . . . . . . . . . . . . . . . . . . p. 70
2.4. Label Correcting Methods . . . . . . . . . . . . . . . . p. 73
2.4.1. The Bellman-Ford Method . . . . . . . . . . . . . . p. 73
2.4.2. The D’Esopo-Pape Algorithm . . . . . . . . . . . . p. 75
2.4.3. The SLF and LLL Algorithms . . . . . . . . . . . . p. 76
2.4.4. The Threshold Algorithm . . . . . . . . . . . . . . p. 78
2.4.5. Comparison of Label Setting and Label Correcting . . . p. 80
2.5. Single Origin/Single Destination Methods . . . . . . . . . p. 81
2.5.1. Label Setting . . . . . . . . . . . . . . . . . . . . p. 81
v
vi Contents
2.5.2. Label Correcting . . . . . . . . . . . . . . . . . . p. 81
2.6. Auction Algorithms . . . . . . . . . . . . . . . . . . . p. 86
2.7. Multiple Origin/Multiple Destination Methods . . . . . . . p. 96
2.8. Notes, Sources, and Exercises . . . . . . . . . . . . . . . p. 98
3. The Max-Flow Problem . . . . . . . . . . . . . . p. 115
3.1. The Max-Flow and Min-Cut Problems . . . . . . . . . . p. 116
3.1.1. Cuts in a Graph . . . . . . . . . . . . . . . . . p. 119
3.1.2. The Max-Flow/Min-Cut Theorem . . . . . . . . . . p. 121
3.1.3. The Maximal and Minimal Saturated Cuts . . . . . . p. 123
3.1.4. Decomposition of Infeasible Network Problems . . . . p. 124
3.2. The Ford-Fulkerson Algorithm . . . . . . . . . . . . . p. 125
3.3. Price-Based Augmenting Path Algorithms . . . . . . . . p. 132
3.3.1. A Price-Based Path Construction Algorithm . . . . . p. 135
3.3.2. A Price-Based Max-Flow Algorithm . . . . . . . . . p. 139
3.4. Notes, Sources, and Exercises . . . . . . . . . . . . . . p. 139
4. The Min-Cost Flow Problem . . . . . . . . . . . . p. 151
4.1. Transformations and Equivalences . . . . . . . . . . . . p. 152
4.1.1. Setting the Lower Flow Bounds to Zero . . . . . . . p. 152
4.1.2. Eliminating the Upper Flow Bounds . . . . . . . . . p. 153
4.1.3. Reduction to a Circulation Format . . . . . . . . . p. 154
4.1.4. Reduction to an Assignment Problem . . . . . . . . p. 154
4.2. Duality . . . . . . . . . . . . . . . . . . . . . . . p. 155
4.2.1. Interpretation of CS and the Dual Problem . . . . . . p. 162
4.2.2. Duality and CS for Nonnegativity Constraints . . . . p. 163
4.3. Notes, Sources, and Exercises . . . . . . . . . . . . . . p. 164
5. Simplex Methods for Min-Cost Flow . . . . . . . . . p. 169
5.1. Main Ideas in Simplex Methods . . . . . . . . . . . . . p. 170
5.1.1. Using Prices to Obtain the In-Arc . . . . . . . . . . p. 176
5.1.2. Obtaining the Out-Arc . . . . . . . . . . . . . . p. 179
5.1.3. Dealing with Degeneracy . . . . . . . . . . . . . . p. 183
5.2. The Basic Simplex Algorithm . . . . . . . . . . . . . . p. 186
5.2.1. Termination Properties of the Simplex Method . . . . p. 187
5.2.2. Initialization of the Simplex Method . . . . . . . . . p. 188
5.3. Extension to Problems with Upper and Lower Bounds . . . p. 195
5.4. Implementation Issues . . . . . . . . . . . . . . . . . p. 199
5.5. Notes, Sources, and Exercises . . . . . . . . . . . . . . p. 203
6. Dual Ascent Methods for Min-Cost Flow . . . . . . . p. 213
6.1. Dual Ascent . . . . . . . . . . . . . . . . . . . . . p. 214
Contents vii
6.2. The Primal-Dual (Sequential Shortest Path) Method . . . p. 221
6.3. The Relaxation Method . . . . . . . . . . . . . . . . p. 234
6.4. Solving Variants of an Already Solved Problem . . . . . . p. 243
6.5. Implementation Issues . . . . . . . . . . . . . . . . . p. 243
6.6. Notes, Sources, and Exercises . . . . . . . . . . . . . . p. 244
7. Auction Algorithms for Min-Cost Flow . . . . . . . p. 251
7.1. The Auction Algorithm for the Assignment Problem . . . p. 252
7.1.1. The Main Auction Algorithm . . . . . . . . . . . . p. 253
7.1.2. Approximate Coordinate Descent Interpretation . . . p. 257
7.1.3. Variants of the Auction Algorithm . . . . . . . . . p. 257
7.1.4. Computational Complexity – (cid:1)-Scaling . . . . . . . . p. 259
7.1.5. Dealing with Infeasibility . . . . . . . . . . . . . p. 265
7.2. Extensions of the Auction Algorithm . . . . . . . . . . p. 268
7.2.1. Reverse Auction . . . . . . . . . . . . . . . . . p. 268
7.2.2. Auction Algorithms for Asymmetric Assignment . . . p. 272
7.2.3. Auction Algorithms with Similar Persons . . . . . . p. 279
7.3. The Preflow-Push Algorithm for Max-Flow . . . . . . . . p. 282
7.3.1. Analysis and Complexity . . . . . . . . . . . . . . p. 285
7.3.2. Implementation Issues . . . . . . . . . . . . . . . p. 293
7.3.3. Relation to the Auction Algorithm . . . . . . . . . p. 294
7.4. The (cid:1)-Relaxation Method . . . . . . . . . . . . . . . p. 304
7.4.1. Computational Complexity – (cid:1)-Scaling . . . . . . . . p. 310
7.4.2. Implementation Issues . . . . . . . . . . . . . . . p. 318
7.5. The Auction/Sequential Shortest Path Algorithm . . . . . p. 320
7.6. Notes, Sources, and Exercises . . . . . . . . . . . . . . p. 326
8. Nonlinear Network Optimization . . . . . . . . . . p. 337
8.1. Convex and Separable Problems . . . . . . . . . . . . p. 339
8.2. Problems with Side Constraints . . . . . . . . . . . . . p. 346
8.3. Multicommodity Flow Problems . . . . . . . . . . . . p. 349
8.4. Integer Constraints . . . . . . . . . . . . . . . . . . p. 355
8.5. Networks with Gains . . . . . . . . . . . . . . . . . p. 360
8.6. Optimality Conditions . . . . . . . . . . . . . . . . . p. 365
8.7. Duality . . . . . . . . . . . . . . . . . . . . . . . p. 370
8.8. Algorithms and Approximations . . . . . . . . . . . . p. 375
8.8.1. Feasible Direction Methods . . . . . . . . . . . . . p. 375
8.8.2. Piecewise Linear Approximation . . . . . . . . . . p. 380
8.8.3. Interior Point Methods . . . . . . . . . . . . . . p. 382
8.8.4. Penalty and Augmented Lagrangian Methods . . . . . p. 384
8.8.5. Proximal Minimization . . . . . . . . . . . . . . p. 386
8.8.6. Smoothing . . . . . . . . . . . . . . . . . . . . p. 387
8.8.7. Transformations . . . . . . . . . . . . . . . . . p. 389
viii Contents
8.9. Notes, Sources, and Exercises . . . . . . . . . . . . . . p. 398
9. Convex Separable Network Problems . . . . . . . . p. 407
9.1. Convex Functions of a Single Variable . . . . . . . . . . p. 408
9.2. Optimality Conditions . . . . . . . . . . . . . . . . . p. 412
9.3. Duality . . . . . . . . . . . . . . . . . . . . . . . p. 414
9.4. Dual Function Differentiability . . . . . . . . . . . . . p. 426
9.5. Algorithms for Differentiable Dual Problems . . . . . . . p. 430
9.6. Auction Algorithms . . . . . . . . . . . . . . . . . . p. 433
9.6.1. The (cid:1)-Relaxation Method . . . . . . . . . . . . . p. 441
9.6.2. Auction/Sequential Shortest Path Algorithm . . . . . p. 446
9.7. Monotropic Programming . . . . . . . . . . . . . . . p. 449
9.8. Notes, Sources, and Exercises . . . . . . . . . . . . . . p. 463
10. Network Problems with Integer Constraints . . . . . p. 467
10.1. Formulation of Integer-Constrained Problems . . . . . . p. 469
10.2. Branch-and-Bound . . . . . . . . . . . . . . . . . . p. 483
10.3. Lagrangian Relaxation . . . . . . . . . . . . . . . . p. 492
10.3.1. Subgradients of the Dual Function . . . . . . . . . p. 497
10.3.2. Subgradient Methods . . . . . . . . . . . . . . . p. 499
10.3.3. Cutting Plane Methods . . . . . . . . . . . . . . p. 503
10.3.4. Decomposition and Multicommodity Flows . . . . . p. 507
10.4. Local Search Methods . . . . . . . . . . . . . . . . p. 512
10.4.1. Genetic Algorithms . . . . . . . . . . . . . . . p. 514
10.4.2. Tabu Search . . . . . . . . . . . . . . . . . . . p. 515
10.4.3. Simulated Annealing . . . . . . . . . . . . . . . p. 516
10.5. Rollout Algorithms . . . . . . . . . . . . . . . . . . p. 517
10.6. Notes, Sources, and Exercises . . . . . . . . . . . . . p. 525
Appendix A: Mathematical Review . . . . . . . . . . p. 545
A.1. Sets . . . . . . . . . . . . . . . . . . . . . . . . p. 546
A.2. Euclidean Space . . . . . . . . . . . . . . . . . . . p. 547
A.3. Matrices . . . . . . . . . . . . . . . . . . . . . . p. 547
A.4. Analysis . . . . . . . . . . . . . . . . . . . . . . . p. 548
A.5. Convex Sets and Functions . . . . . . . . . . . . . . p. 551
A.6. Subgradients . . . . . . . . . . . . . . . . . . . . . p. 553
References . . . . . . . . . . . . . . . . . . . . . . . . . p. 555
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 587
Preface
Network optimization lies in the middle of the great divide that separates
the two major types of optimization problems, continuous and discrete.
The ties between linear programming and combinatorial optimization can
be traced to the representation of the constraint polyhedron as the convex
hull of its extreme points. When a network is involved, however, these ties
becomemuchstrongerbecausetheextremepointsofthepolyhedronarein-
tegerandrepresentsolutionsofcombinatorialproblemsthatareseemingly
unrelated to linear programming. Because of this structure and also be-
causeoftheirintuitivecharacter, networkmodelsprovideidealvehiclesfor
explaining many of the fundamental ideas in both continuous and discrete
optimization.
Aside from their interesting methodological characteristics, network
models are also used extensively in practice, in an ever expanding spec-
trum of applications. Indeed collectively, network problems such as short-
est path, assignment, max-flow, transportation, transhipment, spanning
tree, matching, traveling salesman, generalized assignment, vehicle rout-
ing, and multicommodity flow constitute the most common class of practi-
cal optimization problems. There has been steady progress in the solution
methodology of network problems, and in fact the progress has accelerated
in the last fifteen years thanks to algorithmic and technological advances.
Thepurposeofthisbookistoprovideafairlycomprehensiveandup-
to-datedevelopmentoflinear,nonlinear,anddiscretenetworkoptimization
problems. The interplay between continuous and discrete structures has
beenhighlighted,theassociatedanalyticalandalgorithmicissueshavebeen
treated quite extensively, and a guide to important network models and
applications has been provided.
Regarding continuous network optimization, we focus on two ideas,
which are also fundamental in general mathematical programming: dual-
ity and iterative cost improvement. We provide an extensive treatment of
iterative algorithms for the most common linear cost problem, the mini-
mumcostflowortranshipmentproblem,andforitsconvexcostextensions.
The discussion of duality is comprehensive: it starts with linear network
ix