Linear Programming: Foundations and Extensions

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Автор(ы):Vanderbei Robert J.
06.10.2007
Год изд.:2001
Описание: This book is about constrained optimization. It begins with a thorough treatment of linear programming and proceeds to convex analysis, network flows, integer programming, quadratic programming, and convex optimization. Along the way, dynamic programming and the linear complementarity problem are touched on as well. The book aims to be a first introduction to the subject. Specific examples and concrete algorithms precede more abstract topics. Nevertheless, topics covered are developed in some depth, a large number of numerical examples are worked out in detail, and many recent topics are included, most notably interior-point methods. The exercises at the end of each chapter both illustrate the theory and, in some cases, extend it.
Оглавление:Part 1. Basic Theory—The Simplex Method and Duality [1]
Chapter 1. Introduction [3]
  1. Managing a Production Facility [3]
  2. The Linear Programming Problem [6]
    Exercises [8]
    Notes [10]
Chapter 2. The Simplex Method [13]
  1. An Example [13]
  2. The Simplex Method [16]
  3. Initialization [19]
  4. Unboundedness [22]
  5. Geometry [22]
    Exercises [24]
    Notes [27]
Chapter 3. Degeneracy [29]
  1. Definition of Degeneracy [29]
  2. Two Examples of Degenerate Problems [29]
  3. The Perturbation/Lexicographic Method [32]
  4. Eland's Rule [36]
  5. Fundamental Theorem of Linear Programming [38]
  6. Geometry [39]
    Exercises [42]
    Notes [43]
Chapter 4. Efficiency of the Simplex Method [45]
  1. Performance Measures [45]
  2. Measuring the Size of a Problem [45]
  3. Measuring the Effort to Solve a Problem [46]
  4. Worst-Case Analysis of the Simplex Method [47]
    Exercises [52]
    Notes [53]
Chapter 5. Duality Theory [55]
  1. Motivation—Finding Upper Bounds [55]
  2. The Dual Problem [57]
  3. The Weak Duality Theorem [58]
  4. The Strong Duality Theorem [60]
  5. Complementary Slackness [66]
  6. The Dual Simplex Method [68]
  7. A Dual-Based Phase I Algorithm [71]
  8. The Dual of a Problem in General Form [73]
  9. Resource Allocation Problems [74]
  10. Lagrangian Duality [78]
    Exercises [79]
    Notes [87]
Chapter 6. The Simplex Method in Matrix Notation [89]
  1. Matrix Notation [89]
  2. The Primal Simplex Method [91]
  3. An Example [96]
  4. The Dual Simplex Method [101]
  5. Two-Phase Methods [104]
  6. Negative Transpose Property [105]
    Exercises [108]
    Notes [109]
Chapter 7. Sensitivity and Parametric Analyses [111]
  1. Sensitivity Analysis [111]
  2. Parametric Analysis and the Homotopy Method [115]
  3. The Parametric Self-Dual Simplex Method [119]
    Exercises [120]
    Notes [124]
Chapter 8. Implementation Issues [125]
  1. Solving Systems of Equations: LU-Factorization [126]
  2. Exploiting Sparsity [130]
  3. Reusing a Factorization [136]
  4. Performance Tradeoffs [140]
  5. Updating a Factorization [141]
  6. Shrinking the Bump [145]
  7. Partial Pricing [146]
  8. Steepest Edge [147]
    Exercises [149]
    Notes [150]
Chapter 9. Problems in General Form [151]
  1. The Primal Simplex Method [151]
  2. The Dual Simplex Method [153]
    Exercises [159]
    Notes [160]
Chapter 10. Convex Analysis [161]
  1. Convex Sets [161]
  2. Caratheodory's Theorem [163]
  3. The Separation Theorem [165]
  4. Parkas' Lemma [167]
  5. Strict Complementarity [168]
    Exercises [170]
    Notes [171]
Chapter 11. Game Theory [173]
  1. Matrix Games [173]
  2. Optimal Strategies [175]
  3. The Minimax Theorem [177]
  4. Poker [181]
    Exercises [184]
    Notes [187]
Chapter 12. Regression [189]
  1. Measures of Mediocrity [189]
  2. Multidimensional Measures: Regression Analysis [191]
  3. L(?)-Regression [193]
  4. L(?) -Regression [195]
  5. Iteratively Reweighted Least Squares [196]
  6. An Example: How Fast is the Simplex Method? [198]
  7. Which Variant of the Simplex Method is Best? [202]
    Exercises [203]
    Notes [208]
Part 2. Network-Type Problems [211]
Chapter 13. Network Flow Problems [213]
  1. Networks [213]
  2. Spanning Trees and Bases [216]
  3. The Primal Network Simplex Method [221]
  4. The Dual Network Simplex Method [225]
  5. Putting It All Together [228]
  6. The Integrality Theorem [231]
    Exercises [232]
    Notes [240]
Chapter 14. Applications [241]
  1. The Transportation Problem [241]
  2. The Assignment Problem [243]
  3. The Shortest-Path Problem [244]
  4. Upper-Bounded Network Flow Problems [247]
  5. The Maximum-Flow Problem [250]
    Exercises [252]
    Notes [257]
Chapter 15. Structural Optimization [259]
  1. An Example [259]
  2. Incidence Matrices [261]
  3. Stability [262]
  4. Conservation Laws [264]
  5. Minimum-Weight Structural Design [267]
  6. Anchors Away [269]
    Exercises [272]
    Notes [272]
Part3. Interior-Point Methods [275]
Chapter 16. The Central Path [277]
    Warning: Nonstandard Notation Ahead [277]
  1. The Barrier Problem [277]
  2. Lagrange Multipliers [280]
  3. Lagrange Multipliers Applied to the Barrier Problem [283]
  4. Second-Order Information [285]
  5. Existence [285]
    Exercises [287]
    Notes [289]
Chapter 17. A Path-Following Method [291]
  1. Computing Step Directions [291]
  2. Newton's Method [293]
  3. Estimating an Appropriate Value for the Barrier Parameter [294]
  4. Choosing the Step Length Parameter [295]
  5. Convergence Analysis [296]
    Exercises [302]
    Notes [306]
Chapter 18. The KKT System [307]
  1. The Reduced KKT System [307]
  2. The Normal Equations [308]
  3. Step Direction Decomposition [310]
    Exercises [313]
    Notes [313]
Chapter 19. Implementation Issues [315]
  1. Factoring Positive Definite Matrices [315]
  2. Quasidefinite Matrices [319]
  3. Problems in General Form [325]
    Exercises [331]
    Notes [331]
Chapter 20. The Affine-Scaling Method [333]
  1. The Steepest Ascent Direction [333]
  2. The Projected Gradient Direction [335]
  3. The Projected Gradient Direction with Scaling [337]
  4. Convergence [341]
  5. Feasibility Direction [343]
  6. Problems in Standard Form [344]
    Exercises [345]
    Notes [346]
Chapter 21. The Homogeneous Self-Dual Method [349]
  1. From Standard Form to Self-Dual Form [349]
  2. Homogeneous Self-Dual Problems [350]
  3. Back to Standard Form [360]
  4. Simplex Method vs Interior-Point Methods [363]
    Exercises [367]
    Notes [368]
Part 4. Extensions [371]
Chapter 22. Integer Programming [373]
  1. Scheduling Problems [373]
  2. The Traveling Salesman Problem [375]
  3. Fixed Costs [378]
  4. Nonlinear Objective Functions [378]
  5. Branch-and-Bound [380]
    Exercises [392]
    Notes [393]
Chapter 23. Quadratic Programming [395]
  1. The Markowitz Model [395]
  2. The Dual [399]
  3. Convexity and Complexity [402]
  4. Solution Via Interior-Point Methods [404]
  5. Practical Considerations [406]
    Exercises [409]
    Notes [411]
Chapter 24. Convex Programming [413]
  1. Differentiable Functions and Taylor Approximations [413]
  2. Convex and Concave Functions [414]
  3. Problem Formulation [414]
  4. Solution Via Interior-Point Methods [415]
  5. Successive Quadratic Approximations [417]
  6. Merit Functions [417]
  7. Parting Words [421]
    Exercises [421]
    Notes [423]
Appendix A. Source Listings [425]
  1. The Self-Dual Simplex Method [426]
  2. The Homogeneous Self-Dual Method [429]
Answers to Selected Exercises [433]
Bibliography [435]
Index [443]
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