Linear and Nonlinear Programming (International Series in Operations Research & Management Science, 116)

✍ Scribed by David G. Luenberger, Yinyu Ye

• Publisher: Springer
• Year: 2008
• Tongue: English
• Leaves: 550
• Category: Library

✦ Synopsis

"Linear and Nonlinear Programming" is considered a classic textbook in Optimization. While it is a classic, it also reflects modern theoretical insights. These insights provide structure to what might otherwise be simply a collection of techniques and results, and this is valuable both as a means for learning existing material and for developing new results. One major insight of this type is the connection between the purely analytical character of an optimization problem, expressed perhaps by properties of the necessary conditions, and the behavior of algorithms used to solve a problem. This was a major theme of the first and second editions. Now the third edition has been completely updated with recent Optimization Methods. The new co-author, Yinyu Ye, has written chapters and chapter material on a number of these areas including Interior Point Methods.

✦ Table of Contents

Cover
Half-Title
Title
Copyright
Dedication
Preface
Contents
Chapter 1. Introduction
1.1. Optimization
1.2. Types of Problems
1.3. Size of Problems
1.4. Iterative Algorithms and Convergence
PART I Linear Programming
Chapter 2. Basic Properties of Linear Programs
2.1. Introduction
2.2. Examples of Linear Programming Problems
2.3. Basic Solutions
2.4. The Fundamental Theorem of Linear Programming
2.5. Relations to Convexity
2.6. Exercises
Chapter 3. The Simplex Method
3.1. Pivots
3.2. Adjacent Extreme Points
3.3. Determining a Minimum Feasible Solution
3.4. Computational Procedure—Simplex Method
3.5. Artificial Variables
3.6. Matrix Form of the Simplex Method
3.7. The Revised Simplex Method
∗3.8. The Simplex Method and LU Decomposition
3.9. Decomposition
3.10. Summary
3.11. Exercises
Chapter 4. Duality
4.1. Dual Linear Programs
4.2. The Duality Theorem
4.3. Relations to the Simplex Procedure
4.4. Sensitivity and Complementary Slackness
∗4.5. The Dual Simplex Method
∗4.6. The Primal–Dual Algorithm
∗4.7. Reduction of Linear Inequalities
4.8. Exercises
Chapter 5. Interior-Point Methods
5.1. Elements of Complexity Theory
∗5.2. The Simplex Method is not Polynomial-Time
∗5.3. The Ellipsoid Method
5.4. The Analytic Center
5.5. The Central Path
5.6. Solution Strategies
5.7. Termination and Initialization
5.8. Summary
5.9. Exercises
Chapter 6. Transportation and Network Flow Problems
6.1. The Transportation Problem
6.2. Finding a Basic Feasible Solution
6.3. Basis Triangularity
6.4. Simplex Method for Transportation Problems
6.5. The Assignment Problem
6.6. Basic Network Concepts
6.7. Minimum Cost Flow
6.8. Maximal Flow
6.9. Summary
6.10. Exercises
PART II Unconstrained Problems
Chapter 7. Basic Properties of Solutions and Algorithms
7.1. First-Order Necessary Conditions
7.2. Examples of Unconstrained Problems
7.3. Second-Order Conditions
7.4. Convex and Concave Functions
7.5. Minimization and Maximization of Convex Functions
7.6. Zero-Order Conditions
7.7. Global Convergence of Descent Algorithms
7.8. Speed of Convergence
7.9. Summary
7.10. Exercises
Chapter 8. Basic Descent Methods
8.1. Fibonacci and Golden Section Search
8.2. Line Search by Curve Fitting
8.3. Global Convergence of Curve Fitting
8.4. Closedness of Line Search Algorithms
8.5. Inaccurate Line Search
8.6. The Method of Steepest Descent
8.7. Applications of the Theory
8.8. Newton’s Method
8.9. Coordinate Descent Methods
8.10. Spacer Steps
8.11. Summary
8.12. Exercises
Chapter 9. Conjugate Direction Methods
9.1. Conjugate Directions
9.2. Descent Properties of the Conjugate Direction Method
9.3. The Conjugate Gradient Method
9.4. The C–G Method as an Optimal Process
9.5. The Partial Conjugate Gradient Method
9.6. Extension to Nonquadratic Problems
9.7. Parallel Tangents
9.8. Exercises
Chapter 10. Quasi-Newton Methods
10.1. Modified Newton Method
10.2. Construction of the Inverse
10.3. Davidon–Fletcher–Powell Method
10.4. The Broyden Family
10.5. Convergence Properties
10.6. Scaling
10.7. Memoryless Quasi-Newton Methods
∗10.8. Combination of Steepest Descent and Newton’s Method
10.9. Summary
10.10. Exercises
PART III Constrained Minimization
Chapter 11. Constrained Minimization Conditions
11.1. Constraints
11.2. Tangent Plane
11.3. First-Order Necessary Conditions (Equality Constraints)
11.4. Examples
11.5. Second-Order Conditions
11.6. Eigenvalues in Tangent Subspace
11.7. Sensitivity
11.8. Inequality Constraints
11.9. Zero-Order Conditions and Lagrange Multipliers
11.10. Summary
11.11. Exercises
Chapter 12. Primal Methods
12.1. Advantage of Primal Methods
12.2. Feasible Direction Methods
12.3. Active Set Methods
12.4. The Gradient Projection Method
12.5. Convergence Rate of the Gradient Projection Method
12.6. The Reduced Gradient Method
12.7. Convergence Rate of the Reduced Gradient Method
12.8. Variations
12.9. Summary
12.10. Exercises
Chapter 13. Penalty and Barrier Methods
13.1. Penalty Methods
13.2. Barrier Methods
13.3. Properties of Penalty and Barrier Functions
13.4. Newton’s Method and Penalty Functions
13.5. Conjugate Gradients and Penalty Methods
13.6. Normalization of Penalty Functions
13.7. Penalty Functions and Gradient Projection
13.8. Exact Penalty Functions
13.9. Summary
13.10. Exercises
Chapter 14. Dual and Cutting Plane Methods
14.1. Global Duality
14.2. Local Duality
14.3. Dual Canonical Convergence Rate
14.4. Separable Problems
14.5. Augmented Lagrangians
14.6. The Dual Viewpoint
14.7. Cutting Plane Methods
14.8. Kelley’s Convex Cutting Plane Algorithm
14.9. Modifications
14.10. Exercises
Chapter 15. Primal-Dual Methods
15.1. The Standard Problem
15.2. Strategies
15.3. A Simple Merit Function
15.4. Basic Primal–Dual Methods
15.5. Modified Newton Methods
15.6. Descent Properties
15.7. Rate of Convergence
15.8. Interior Point Methods
15.9. Semidefinite Programming
15.10. Summary
15.11. Exercises
Appendix A. Mathematical Review
A.1. Sets
A.2. Matrix Notation
A.3. Spaces
A.4. Eigenvalues and Quadratic Forms
A.5. Topological Concepts
A.6. Functions
Appendix B. Convex Sets
B.1. Basic Definitions
B.2. Hyperplanes and Polytopes
B.3. Separating and Supporting Hyperplanes
B.4. Extreme Points
Appendix C. Gaussian Elimination
Bibliography
Index