Optimization Theory for Large Systems

Transcription

1 Optimization Theory for Large Systems LEON S. LASDON CASE WESTERN RESERVE UNIVERSITY THE MACMILLAN COMPANY COLLIER-MACMILLAN LIMITED, LONDON

2 Contents 1. Linear and Nonlinear Programming Unconstrained Minimization Linear Programming Simplex Method Revised Simplex Method Duality in Linear Programming Dual Simplex and Primal-Dual Algorithms Nonlinear Programming Convexity Kuhn-Tucker Conditions Saddle Points and Sufficiency Conditions Methods of Nonlinear Programming 91 REFERENCES Large Mathematical Programs with Special Structure Introduction Activity Analysis Production and Inventory Problem Dynamic Leontief Model Angular and Dual-Angular Structures Linear Programs with Many Rows or Columns 122 vii

3 2.7 Nonlinear Programs with Coupling Variables Mixed-Variable Programs and a Location Problem 135 PROBLEMS 142 REFERENCES 143 CONTENTS The Dantzig-Wolfe Decomposition Principle Introduction A Theorem on Conyex Combinations Column Generation Development of the Decomposition Principle Example of the Decomposition Principle Economic Interpretation of the Decomposition Principle Lower Bound for the Minimal Cost Application to Transportation Problems Generalized Transportation Problems and a Forestry-Cutting Example Optimal Allocation of Limited Resources General Formulation Specializing the Model Lot Sizes and Labor Allocations Computational Experience Primal-Dual Approach to the Master Program Linear Fractional Programming Application of the Primal-Dual Method to the Master Program Example of the Primal-Dual Method Three Algorithms for Solving the Master Program A Comparison 201 PROBLEMS 203 REFERENCES 205 Solution of Linear Programs with Many Columns by Column-Generation Procedures The Cutting-Stock Problem Column-Generation and Multi-item Scheduling Generalized Linear Programming Grid Linearization and Nonlinear Programming General Development 242

4 CONTENTS Nonlinear Version of the Dantzig-Wolfe Decomposition Principle Design of Multiterminal Flow Networks 254 PROBLEMS 263 REFERENCES 265 ix 5. Partitioning and Relaxation Procedures in Linear Programming Introduction Relaxation Problems with Coupling Constraints and Coupling Variables Rosen's Partitioning Procedure for Angular and Dual-Angular Problems Development of the Algorithm Computational Considerations Computational Experience Example of Rosen's Partitioning Method 298 PROBLEMS 302 REFERENCES Compact Inverse Methods Introduction Revised Simplex Method with Inverse in Product Form Upper Bounding Methods Generalized Upper Bounding Development of the Algorithm Example of the Generalized Upper Bounding Method Extension to Angular Structures 340 PROBLEMS 356 REFERENCES 356 \ 7. Partitioning Procedures in Nonlinear Programming Introduction Rosen's Partitioning Algorithm for Nonlinear Programs 359

5 X CONTENTS Development of the Algorithm Use of Partition Programming in Refinery Optimization Benders' Partitioning Algorithm for Mixed-Variable Programming Problems Development of the Algorithm Relation to the Decomposition Principle and Cutting-Plane Algorithms Application to a Warehouse Location Problem Numerical Example Computational Experience 389 PROBLEMS 392 REFERENCES 394 Q> Duality and Decomposition in Mathematicai Programming Introduction Decomposition Using a Pricing Mechanism Saddle Points of Lagrangian Functions Basic Theorems Everetts Theorem Application to Linear Integer Programs Minimax Dual Problem Differentiability of the Dual Objective Function Computational Methods for Solving the Dual Special Results for Convex Problems Applications Problems Involving Coupled Subsystems Example Optimal Control of Discrete-Time Dynamic Systems Problems in Which the Constraint Set is Finite: Multi-item Scheduling Problems 449 PROBLEMS 456 REFERENCES Decomposition By Right-Hand-Side Allocation Introduction Problem Formulation Feasible-Directions Algorithm for the Master Program 464

6 CONTENTS 9.4 Alternative Approach to the Direction-Finding Problem 9.5 Tangential Approximation 482 PROBLEMS 491 REFERENCES 491 Appendix 1. Convex Functions and Their Conjugates 493 Appendix 2. Subgradients and Directional Derivatives of Convex Functions 502 REFERENCES 513 List of Symbols 515 Index 517