Decomposition Techniques in Mathematical Programming
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1 Antonio J. Conejo Enrique Castillo Roberto Minguez Raquel Garcia-Bertrand Decomposition Techniques in Mathematical Programming Engineering and Science Applications Springer
2 Contents Part I Motivation and Introduction 1 Motivating Examples Motivation Introduction Linear Programming: Complicating Constraints Transnational Soda Company Stochastic Hydro Scheduling River Basin Operation Energy Production Model Linear Programming: Complicating Variables Two-Year Coal and Gas Procurement Capacity Expansion Planning The Water Supply System Nonlinear Programming: Complicating Constraints Production Scheduling Operation of a Multiarea Electricity Network The Wall Design Reliability-based Optimization of a Rubblemound Breakwater Nonlinear Programming: Complicating Variables Capacity Expansion Planning: Revisited Mixed-Integer Programming: Complicating Constraints Unit Commitment Mixed-Integer Programming: Complicating Variables Capacity Expansion Planning: Revisited The Water Supply System: Revisited Concluding Remarks Exercises 62
3 XII Contents Part II Decomposition Techniques 2 Linear Programming: Complicating Constraints Introduction Complicating Constraints: Problem Structure Decomposition The Dantzig-Wolfe Decomposition Algorithm Description Bounds Issues Related to the Master Problem Alternative Formulation of the Master Problem Concluding Remarks Exercises Linear Programming: Complicating Variables Introduction Complicating Variables: Problem Structure Benders Decomposition Description Bounds The Benders Decomposition Algorithm Subproblem Infeasibility Concluding Remarks Exercises Duality Introduction Karush-Kuhn-Tucker First- and Second-Order Optimality Conditions Equality Constraints and Newton Algorithm Duality in Linear Programming Obtaining the Dual Problem from a Primal Problem in Standard Form Obtaining the Dual Problem Duality Theorems Duality in Nonlinear Programming Illustration of Duality and Separability Concluding Remarks Exercises Decomposition in Nonlinear Programming Introduction Complicating Constraints Lagrangian Relaxation 187
4 Contents XIII Decomposition Algorithm Dual Infeasibility Multiplier Updating Augmented Lagrangian Decomposition Decomposition Algorithm Separability Multiplier Updating Penalty Parameter Updating Optimality Condition Decomposition (OCD) Motivation: Modified Lagrangian Relaxation Decomposition Structure Decomposition Algorithm Convergence Properties Complicating Variables Introduction Benders Decomposition Algorithm From Lagrangian Relaxation to Dantzig-Wolfe Decomposition Lagrangian Relaxation in LP Dantzig-Wolfe from Lagrangian Relaxation Concluding Remarks Exercises Decomposition in Mixed-Integer Programming Introduction Mixed-Integer Linear Programming The Benders Decomposition for MILP Problems Convergence Mixed-Integer Nonlinear Programming Complicating Variables: Nonlinear Case The Benders Decomposition Subproblem Infeasibility Convergence Complicating Constraints: Nonlinear Case Outer Linearization Algorithm Convergence Concluding Remarks Exercises 264
5 XIV Contents 7 Other Decomposition Techniques Bilevel Decomposition A Relaxation Method The Cutting Hyperplane Method Bilevel Programming Equilibrium Problems Coordinate Descent Decomposition Banded Matrix Structure Problems Exercises 297 Part III Local Sensitivity Analysis 8 Local Sensitivity Analysis Introduction Statement of the Problem Sensitivities Based on Duality Theory Karush-Kuhn-Tucker Conditions Obtaining the Set of All Dual Variable Values Some Sensitivities of the Objective Function A Practical Method for the Sensitivities of the Objective Function A General Formula for the Sensitivities of the Objective Function A General Method for Obtaining All Sensitivities Determining the Set of All Feasible Perturbations Discussion of Directional and Partial Derivatives Determining Directional Derivatives if They Exist Partial Derivatives Obtaining All Sensitivities at Once Particular Cases No Constraints Same Active Constraints The General Case Sensitivities of Active Constraints Exercises 341 Part IV Applications 9 Applications The Wall Design Method 1: Updating Safety Factor Bounds Method 2: Using Cutting Planes The Bridge Crane Design 361
6 Contents XV Obtaining Relevant Constraints A Numerical Example Network Constrained Unit Commitment Introduction Notation Problem Formulation Solution Approach Production Costing Introduction Notation Problem Formulation Solution Approach Hydrothermal Coordination Introduction Notation Problem Formulation Solution Approach Multiarea Optimal Power Flow Introduction Notation Problem Formulation Solution Approach Sensitivity in Regression Models 389 Part V Computer Codes A Some GAMS Implementations 397 A.l Dantzig-Wolfe Algorithm 397 A.2 Benders Decomposition Algorithm 403 A.3 GAMS Code for the Rubblemound Breakwater Example 407 A.4 GAMS Code for the Wall Problem 410 A.4.1 The Relaxation Method 410 A.4.2 The Cutting Hyperplanes Method 414 Part VI Solution to Selected Exercises B Exercise Solutions 421 B.l Exercises from Chapter B.2 Exercises from Chapter B.3 Exercises from Chapter B.4 Exercises from Chapter B.5 Exercises from Chapter B.6 Exercises from Chapter 6 475
7 XVI Contents B.7 Exercises from Chapter B.8 Exercises from Chapter References 531 Index 537
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