ISBN: 3540658335
TITLE: Linear Optimization and Extensions
AUTHOR: Padberg, Manfred
TOC:

1. Introduction 1
1.1 Some Issues in Linear Computation 7
1.2 Three Examples of Linear Computation 13
1.2.1 Gargantuan Liquids, Inc. 13
1.2.2 Oil Refineries, bpd 15
1.2.3 Save Berlin, usw. 20
2. The Linear Programming Problem 25
2.1 Standard and Canonical Forms 26
2.2 Matrices, Vectors, Scalars 27
3. Basic Concepts 33
3.1 A Fundamental Theorem 36
3.2 Notational Conventions and Illustrations 39
4. Five Preliminaries 43
4.1 Bases and Basic Feasible Solutions 43
4.2 Detecting Optimality 43
4.3 Detecting Unboundedness 44
4.4 A Rank-One Update 45
4.5 Changing Bases 45
5. Simplex Algorithms 49
5.1 Notation, Reading Instructions, Updating 50
5.2 Big M or How to Get Started 54
5.3 Selecting a Pivot Row and Column 56
5.4 Data Structures, Tolerances, Product Form 58
5.5 Equation Format and Cycling 63
5.6 Finiteness of a Simplex Algorithm 69
5.7 Canonical Form 71
5.7.1 A Worst-Case Example for a Simplex Algorithm 75
5.8 Block Pivots and Structure 77
5.8.1 A Generalized Product Form 79
5.8.2 Upper Bounds 82
6. Primal-Dual Pairs 87
6.1 Weak Duality 89
6.2 Strong Duality 91
6.2.1 Economic Interpretation and Applications 94
6.3 Solvability, Redundancy, Separability 97
6.4 A Dual Simplex Algorithm 103
6.4.1 Correctness, Finiteness, Initialization 105
6.5 Post-Optimality 109
6.6 A Dynamic Simplex Algorithm 114
7. Analytical Geometry 121
7.1 Points, Lines, Subspaces 124
7.2 Polyhedra, Ideal Descriptions, Cones 131
7.2.1 Faces, Valid Equations, Affine Hulls 134
7.2.2 Facets, Minimal Complete Descriptions, Quasi-Uniqueness 138
7.2.3 Asymptotic Cones and Extreme Rays 141
7.2.4 Adjacency I, Extreme Rays of Polyhedra, Homogenization 144
7.3 Point Sets, Affine Transformations, Minimal Generators 147
7.3.1 Displaced Cones, Adjacency II, Images of Polyhedra 150
7.3.2 Carathodory, Minkowski, Weyl 155
7.3.3 Minimal Generators, Canonical Generators, Quasi-Uniqueness 157
7.4 Double Description Algorithms 165
7.4.1 Correctness and Finiteness of the Algorithm 168
7.4.2 Geometry, Euclidean Reduction, Analysis 173
7.4.3 The Basis Algorithm and All-Integer Inversion 180
7.4.4 An All-Integer Algorithm for Double Description 183
7.5 Digital Sizes of Rational Polyhedra and Linear Optimization 188
7.5.1 Facet Complexity, Vertex Complexity, Complexity of Inversion 190
7.5.2 Polyhedra and Related Polytopes for Linear Optimization 194
7.5.3 Feasibility, Binary Search, Linear Optimization 197
7.5.4 Perturbation, Uniqueness, Separation 202
7.6 Geometry and Complexity of Simplex Algorithms 207
7.6.1 Pivot Column Choice, Simplex Paths, Big M Revisited 208
7.6.2 Gaussian Elimination, Fill-In, Scaling 212
7.6.3 Iterative Step I, Pivot Choice, Cholesky Factorization. 216
7.6.4 Cross Multiplication, Iterative Step II, Integer Factorization 219
7.6.5 Division Free Gaussian Elimination and Cramer's Rule 221
7.7 Circles, Spheres, Ellipsoids 229
8. Projective Algorithms 239
8.1 A Basic Algorithm 243
8.1.1 The Solution of the Approximate Problem 245
8.1.2 Convergence of the Approximate Iterates 246
8.1.3 Correctness, Finiteness, Initialization 250
8.2 Analysis, Algebra, Geometry 253
8.2.1 Solution to the Problem in the Original Space 254
8.2.2 The Solution in the Transformed Space 260
8.2.3 Geometric Interpretations and Properties 264
8.2.4 Extending the Exact Solution and Proofs 268
8.2.5 Examples of Projective Images 271
8.3 The Cross Ratio 274
8.4 Reflection on a Circle and Sandwiching 278
8.4.1 The Iterative Step 283
8.5 A Projective Algorithm 288
8.6 Centers, Barriers, Newton Steps 292
8.6.1 A Method of Centers 296
8.6.2 The Logarithmic Barrier Function 298
8.6.3 A Newtonian Algorithm 303
8.7 Coda 308
9. Ellipsoid Algorithms 309
9.1 Matrix Norms, Approximate Inverses, Matrix Inequalities 316
9.2 Ellipsoid "Halving" in Approximate Arithmetic 320
9.3 Polynomial-Time Algorithms for Linear Programming 328
9.3.1 Linear Programming and Binary Search 336
9.4 Deep Cuts, Sliding Objective, Large Steps, Line Search 339

9.4.1 Linear Programming the Ellipsoidal Way: Two Examples 344
9.4.2 Correctness and Finiteness of the DCS Ellipsoid Algorithm 348
9.5 Optimal Separators, Most Violated Separators, Separation 352
9.6 epsilon-Solidification of Flats, Polytopal Norms, Rounding 356
9.6.1 Rational Rounding and Continued Fractions 361
9.7 Optimization and Separation 368
9.7.1 epsilon-Optimal Sets and epsilon-Optimal Solutions 371
9.7.2 Finding Direction Vectors in the Asymptotic Cone 373
9.7.3 A CCS Ellipsoid Algorithm 375
9.7.4 Linear Optimization and Polyhedral Separation 378
10. Combinatorial Optimization: An Introduction 387
10.1 The Berlin Airlift Model Revisited 389
10.2 Complete Formulations and Their Implications 396
10.3 Extremal Characterizations of Ideal Formulations 405
10.3.1 Blocking and Antiblocking Polyhedra 414
10.4 Polyhedra with the Integrality Property 417
Appendices
A. Short-Term Financial Management 423
B. Operations Management in a Refinery 427
C. Automatized Production: PCBs and Ulysses' Problem 441
References 457
Bibliography 479
Index 495
END
