Optimization Theory for Large Systems,9780486419992

Optimization Theory for Large Systems

by
ISBN13: 9780486419992
ISBN10: 0486419991
Format: Paperback
Pub. Date: 2011-12-28
Publisher(s): Dover Publications
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Summary

Important text examines most significant algorithms for optimizing large systems and clarifying relations between optimization procedures. Much data appear as charts and graphs and will be highly valuable to readers in selecting a method and estimating computer time and cost in problem-solving. Initial chapter on linear and nonlinear programming presents all necessary background for subjects covered in rest of book. Second chapter illustrates how large-scale mathematical programs arise from real-world problems. Appendixes. List of Symbols.

Table of Contents

Linear and Nonlinear Programming
1(103)
Unconstrained Minimization
1(19)
Linear Programming
20(44)
Simplex Method
20(20)
Revised Simplex Method
40(6)
Duality in Linear Programming
46(6)
Dual Simplex and Primal--Dual Algorithms
52(12)
Nonlinear Programming
64(40)
Convexity
68(5)
Kuhn--Tucker Conditions
73(10)
Saddle Points and Sufficiency Conditions
83(8)
Methods of Nonlinear Programming
91(11)
References
102(2)
Large Mathematical Programs with Special Structure
104(40)
Introduction
104(2)
Activity Analysis
106(3)
Production and Inventory Problem
109(1)
Dynamic Leontief Model
110(7)
Angular and Dual-Angular Structures
117(5)
Linear Programs with Many Rows or Columns
122(8)
Nonlinear Programs with Coupling Variables
130(5)
Mixed-Variable Programs and a Location Problem
135(9)
Problems
142(1)
References
143(1)
The Dentzig--Wolfe Decomposition Principle
144(63)
Introduction
144(1)
A Theorem on Convex Combinations
145(1)
Column Generation
146(2)
Development of the Decomposition Principle
148(7)
Example of the Decomposition Principle
155(5)
Economic Interpretation of the Decomposition Principle
160(3)
Lower Bound for the Minimal Cost
163(2)
Application to Transportation Problems
165(3)
Generalized Transportation Problems and a Forestry-Cutting Example
168(3)
Optimal Allocation of Limited Resources
171(14)
General Formulation
171
Specializing the Model---Lot Sizes and Labor Allocations
117(64)
Computational Experience
181(4)
Primal--Dual Approach to the Master Program
185(16)
Linear Fractional Programming
185(8)
Application of the Primal--Dual Method to the Master Program
193(4)
Example of the Primal--Dual Method
197(4)
Three Algorithms for Solving the Master Program---A Comparison
201(6)
Problems
203(2)
References
205(2)
Solution of Linear Programs with Many Columns by Column-Generation Procedures
207(60)
The Cutting-Stock Problem
207(10)
Column-Generation and Multi-item Scheduling
217(13)
Generalized Linear Programming
230(12)
Grid Linearization and Nonlinear Programming
242(12)
General Development
242(10)
Nonlinear Version of the Dantzing--Wolfe Decomposition Principle
252(2)
Design of Multiterminal Flow Networks
254(13)
Problems
263(2)
References
265(2)
Partitioning and Relaxation Procedures in Linear Programming
267(37)
Introduction
267(1)
Relaxation
268(8)
Problems with Coupling Constraints and Coupling Variables
276(8)
Rosen's Partitioning Procedure for Angular and Dual-Angular Problems
284(20)
Development of the Algorithm
284(7)
Computational Considerations
291(5)
Computational Experience
296(2)
Example of Rosen's Partitioning Method
298(4)
Problems
302(1)
References
303(1)
Compact Inverse Methods
304(54)
Introduction
304(1)
Revised Simplex Method with Inverse in Product Form
304(15)
Upper Bounding Methods
319(5)
Generalized Upper Bounding
324(16)
Development of the Algorithm
324(10)
Example of the Generalized Upper Bounding Method
334(6)
Extension to Angular Structures
340(18)
Problems
356(1)
References
356(2)
Partitioning Procedures in Nonlinear Programming
358(38)
Introduction
358(1)
Rosen's Partitioning Algorithm for Nonlinear Programs
359(11)
Development of the Algorithm
360(9)
Use of Partition Programming in Refinery Optimization
369(1)
Benders' Partitioning Algorithm for Mixed-Variable Programming Problems
370(26)
Development of the Algorithm
370(11)
Relation to the Decomposition Principle and Cutting-Plane Algorithms
381(4)
Application to a Warehouse Location Problem
385(3)
Numerical Example
388(1)
Computational Experience
389(3)
Problems
392(2)
References
394(2)
Duality and Decomposition in Mathematical Programming
396(64)
Introduction
396(1)
Decomposition Using a Pricing Mechanism
397(2)
Saddle Points of Lagrangian Functions
399(7)
Basic Theorems
399(3)
Everetts Theorem
402(2)
Application to Linear Integer Programs
404(2)
Minimax Dual Problem
406(13)
Differentiability of the Dual Objective Function
419(9)
Computational Methods for Solving the Dual
428(7)
Special Results for Convex Problems
435(5)
Applications
440(20)
Problems Involving Coupled Subsystems
440(6)
Example---Optimal Control of Discrete-Time Dynamic Systems
446(3)
Problems in Which the Constraint Set is Finite: Multi-item Scheduling Problems
449(7)
Problems
456(2)
References
458(2)
Decomposition By Right-Hand-Side Allocation
460(33)
Introduction
460(1)
Problem Formulation
460(4)
Feasible-Directions Algorithm for the Master Program
464(11)
Alternative Approach to the Direction-Finding Problem
475(7)
Tangential Approximation
482(11)
Problems
491(1)
References
491(2)
Appendix 1. Convex Functions and Their Conjugates 493(9)
Appendix 2. Subgradients and Directional Derivatives of Convex Functions 502(13)
References
513(2)
List of Symbols 515(2)
Index 517

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