Decomposition Techniques in Mathematical Programming,9783540276852

Decomposition Techniques in Mathematical Programming

by ; ; ;
ISBN13: 9783540276852
ISBN10: 3540276858
Format: Hardcover
Pub. Date: 2006-04-30
Publisher(s): Springer Verlag
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Summary

This textbook for students and practitioners presents a practical approach to decomposition techniques in optimization. It provides an appropriate blend of theoretical background and practical applications in engineering and science, which makes the book interesting for practitioners, as well as engineering, operations research and applied economics graduate and postgraduate students. "Decomposition Techniques in Mathematical Programming" is based on clarifying, illustrative and computational examples and applications from electrical, mechanical, energy and civil engineering as well as applied mathematics and economics. It addresses decomposition in linear programming, mixed-integer linear programming, nonlinear programming, and mixed-integer nonlinear programming, and provides rigorous decomposition algorithms as well as heuristic ones. Practical applications are developed up to working algorithms that can be readily used. The theoretical background of the book is deep enough to be of interest to applied mathematicians. It includes end of chapter exercises and the solutions of the even numbered exercises are included as an appendix.

Table of Contents

Part I Motivation and Introduction
1 Motivating Examples
3(64)
1.1 Motivation
3(4)
1.2 Introduction
7(1)
1.3 Linear Programming: Complicating Constraints
8(20)
1.3.1 Transnational Soda Company
8(4)
1.3.2 Stochastic Hydro Scheduling
12(7)
1.3.3 River Basin Operation
19(4)
1.3.4 Energy Production Model
23(5)
1.4 Linear Programming: Complicating Variables
28(11)
1.4.1 Two-Year Coal and Gas Procurement
28(4)
1.4.2 Capacity Expansion Planning
32(4)
1.4.3 The Water Supply System
36(3)
1.5 Nonlinear Programming: Complicating Constraints
39(14)
1.5.1 Production Scheduling
39(3)
1.5.2 Operation of a Multiarea Electricity Network
42(3)
1.5.3 The Wall Design
45(3)
1.5.4 Reliability-based Optimization of a Rubblemound Breakwater
48(5)
1.6 Nonlinear Programming: Complicating Variables
53(2)
1.6.1 Capacity Expansion Planning: Revisited
53(2)
1.7 Mixed-Integer Programming: Complicating Constraints
55(2)
1.7.1 Unit Commitment
55(2)
1.8 Mixed-Integer Programming: Complicating Variables
57(4)
1.8.1 Capacity Expansion Planning: Revisited 2
57(3)
1.8.2 The Water Supply System: Revisited
60(1)
1.9 Concluding Remarks
61(1)
1.10 Exercises
62(5)
Part II Decomposition Techniques
2 Linear Programming: Complicating Constraints
67(40)
2.1 Introduction
67(3)
2.2 Complicating Constraints: Problem Structure
70(3)
2.3 Decomposition
73(4)
2.4 The Dantzig-Wolfe Decomposition Algorithm
77(22)
2.4.1 Description
77(10)
2.4.2 Bounds
87(1)
2.4.3 Issues Related to the Master Problem
88(5)
2.4.4 Alternative Formulation of the Master Problem
93(6)
2.5 Concluding Remarks
99(1)
2.6 Exercises
100(7)
3 Linear Programming: Complicating Variables
107(34)
3.1 Introduction
107(3)
3.2 Complicating Variables: Problem Structure
110(1)
3.3 Benders Decomposition
111(24)
3.3.1 Description
111(5)
3.3.2 Bounds
116(1)
3.3.3 The Benders Decomposition Algorithm
116(12)
3.3.4 Subproblem Infeasibility
128(7)
3.4 Concluding Remarks
135(1)
3.5 Exercises
136(5)
4 Duality
141(46)
4.1 Introduction
141(1)
4.2 Karush—Kuhn—Tucker First- and Second-Order Optimality Conditions
142(7)
4.2.1 Equality Constraints and Newton Algorithm
147(2)
4.3 Duality in Linear Programming
149(12)
4.3.1 Obtaining the Dual Problem from a Primal Problem in Standard Form
150(1)
4.3.2 Obtaining the Dual Problem
151(3)
4.3.3 Duality Theorems
154(7)
4.4 Duality in Nonlinear Programming
161(15)
4.5 Illustration of Duality and Separability
176(5)
4.6 Concluding Remarks
181(1)
4.7 Exercises
181(6)
5 Decomposition in Nonlinear Programming
187(56)
5.1 Introduction
187(1)
5.2 Complicating Constraints
187(1)
5.3 Lagrangian Relaxation
187(18)
5.3.1 Decomposition
188(6)
5.3.2 Algorithm
194(1)
5.3.3 Dual Infeasibility
195(1)
5.3.4 Multiplier Updating
195(10)
5.4 Augmented Lagrangian Decomposition
205(5)
5.4.1 Decomposition
205(2)
5.4.2 Algorithm
207(1)
5.4.3 Separability
208(1)
5.4.4 Multiplier Updating
208(1)
5.4.5 Penalty Parameter Updating
208(2)
5.5 Optimality Condition Decomposition (OCD)
210(13)
5.5.1 Motivation: Modified Lagrangian Relaxation
211(2)
5.5.2 Decomposition Structure
213(1)
5.5.3 Decomposition
214(2)
5.5.4 Algorithm
216(1)
5.5.5 Convergence Properties
217(6)
5.6 Complicating Variables
223(10)
5.6.1 Introduction
223(1)
5.6.2 Benders Decomposition
223(2)
5.6.3 Algorithm
225(8)
5.7 From Lagrangian Relaxation to Dantzig-Wolfe Decomposition
233(5)
5.7.1 Lagrangian Relaxation in LP
234(2)
5.7.2 Dantzig-Wolfe from Lagrangian Relaxation
236(2)
5.8 Concluding Remarks
238(1)
5.9 Exercises
239(4)
6 Decomposition in Mixed-Integer Programming
243(28)
6.1 Introduction
243(1)
6.2 Mixed-Integer Linear Programming
244(7)
6.2.1 The Benders Decomposition for MILP Problems
245(5)
6.2.2 Convergence
250(1)
6.3 Mixed-Integer Nonlinear Programming
251(1)
6.4 Complicating Variables: Nonlinear Case
251(6)
6.4.1 The Benders Decomposition
251(2)
6.4.2 Subproblem Infeasibility
253(4)
6.4.3 Convergence
257(1)
6.5 Complicating Constraints: Nonlinear Case
257(7)
6.5.1 Outer Linearization Algorithm
258(6)
6.5.2 Convergence
264(1)
6.6 Concluding Remarks
264(1)
6.7 Exercises
264(7)
7 Other Decomposition Techniques
271(32)
7.1 Bilevel Decomposition
271(9)
7.1.1 A Relaxation Method
272(5)
7.1.2 The Cutting Hyperplane Method
277(3)
7.2 Bilevel Programming
280(2)
7.3 Equilibrium Problems
282(3)
7.4 Coordinate Descent Decomposition
285(12)
7.4.1 Banded Matrix Structure Problems
287(10)
7.5 Exercises
297(6)
Part III Local Sensitivity Analysis
8 Local Sensitivity Analysis
303(46)
8.1 Introduction
303(1)
8.2 Statement of the Problem
304(1)
8.3 Sensitivities Based on Duality Theory
305(10)
8.3.1 Karush—Kuhn—Tucker Conditions
305(2)
8.3.2 Obtaining the Set of All Dual Variable Values
307(1)
8.3.3 Some Sensitivities of the Objective Function
308(2)
8.3.4 A Practical Method for the Sensitivities of the Objective Function
310(1)
8.3.5 A General Formula for the Sensitivities of the Objective Function
310(5)
8.4 A General Method for Obtaining All Sensitivities
315(6)
8.4.1 Determining the Set of All Feasible Perturbations
317(1)
8.4.2 Discussion of Directional and Partial Derivatives
318(2)
8.4.3 Determining Directional Derivatives if They Exist
320(1)
8.4.4 Partial Derivatives
320(1)
8.4.5 Obtaining All Sensitivities at Once
321(1)
8.5 Particular Cases
321(18)
8.5.1 No Constraints
321(2)
8.5.2 Same Active Constraints
323(3)
8.5.3 The General Case
326(13)
8.6 Sensitivities of Active Constraints
339(2)
8.7 Exercises
341(8)
Part IV Applications
9 Applications
349(48)
9.1 The Wall Design
349(12)
9.1.1 Method 1: Updating Safety Factor Bounds
355(4)
9.1.2 Method 2: Using Cutting Planes
359(2)
9.2 The Bridge Crane Design
361(7)
9.2.1 Obtaining Relevant Constraints
364(1)
9.2.2 A Numerical Example
365(3)
9.3 Network Constrained Unit Commitment
368(6)
9.3.1 Introduction
368(1)
9.3.2 Notation
369(1)
9.3.3 Problem Formulation
370(1)
9.3.4 Solution Approach
371(3)
9.4 Production Costing
374(7)
9.4.1 Introduction
374(1)
9.4.2 Notation
375(1)
9.4.3 Problem Formulation
376(1)
9.4.4 Solution Approach
377(4)
9.5 Hydrothermal Coordination
381(4)
9.5.1 Introduction
381(1)
9.5.2 Notation
382(1)
9.5.3 Problem Formulation
383(1)
9.5.4 Solution Approach
384(1)
9.6 Multiarea Optimal Power Flow
385(4)
9.6.1 Introduction
385(1)
9.6.2 Notation
386(1)
9.6.3 Problem Formulation
387(2)
9.6.4 Solution Approach
389(1)
9.7 Sensitivity in Regression Models
389(8)
Part V Computer Codes
A Some GAMS Implementations
397(24)
A.1 Dantzig-Wolfe Algorithm
397(6)
A.2 Benders Decomposition Algorithm
403(4)
A.3 GAMS Code for the Rubblemound Breakwater Example
407(3)
A.4 GAMS Code for the Wall Problem
410(11)
A.4.1 The Relaxation Method
410(4)
A.4.2 The Cutting Hyperplanes Method
414
Part VI Solution to Selected Exercises
B Exercise Solutions
421(110)
B.1 Exercises from Chapter 1
421(5)
B.2 Exercises from Chapter 2
426(9)
B.3 Exercises from Chapter 3
435(6)
B.4 Exercises from Chapter 4
441(10)
B.5 Exercises from Chapter 5
451(24)
B.6 Exercises from Chapter 6
475(25)
B.7 Exercises from Chapter 7
500(6)
B.8 Exercises from Chapter 8
506(25)
References 531(6)
Index 537

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