An Introduction to Optimization, 2nd Edition,9780471391265

An Introduction to Optimization, 2nd Edition

by ;
Edition: 2nd
ISBN13: 9780471391265
ISBN10: 0471391263
Format: Hardcover
Pub. Date: 2001-08-01
Publisher(s): Wiley-Interscience
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Summary

An Instructor's Manual presenting detailed solutions to all the problems in the book is available from the Wiley editorial department.

Author Biography

EDWIN K. P. CHONG, PhD, is Professor of Electrical and Computer Engineering at Colorado State University, Fort Collins, Colorado. He was an Associate Editor for the IEEE Transactions on Automatic Control and received the 1998 ASEE Frederick Emmons Terman Award.

Table of Contents

Preface xiii
Part I Mathematical Review
Methods of Proof and Some Notation
1(4)
Methods of Proof
1(2)
Notation
3(2)
Exercises
4(1)
Vector Spaces and Matrices
5(16)
Real Vector Spaces
5(5)
Rank of a Matrix
10(4)
Linear Equations
14(2)
Inner Products and Norms
16(5)
Exercises
19(2)
Transformations
21(18)
Linear Transformations
21(1)
Eigenvalues and Eigenvectors
22(3)
Orthogonal Projections
25(1)
Quadratic Forms
26(5)
Matrix Norms
31(8)
Exercises
35(4)
Concepts from Geometry
39(10)
Line Segments
39(1)
Hyperplanes and Linear Varieties
39(3)
Convex Sets
42(2)
Neighborhoods
44(1)
Polytopes and Polyhedra
45(4)
Exercises
47(2)
Elements of Calculus
49(24)
Sequences and Limits
49(6)
Differentiability
55(2)
The Derivative Matrix
57(2)
Differentiation Rules
59(1)
Level Sets and Gradients
60(4)
Taylor Series
64(9)
Exercises
68(5)
Part II Unconstrained Optimization
Basics of Set-Constrained and Unconstrained Optimization
73(18)
Introduction
73(2)
Conditions for Local Minimizers
75(16)
Exercises
83(8)
One-Dimensional Search Methods
91(22)
Golden Section Search
91(4)
Fibonacci Search
95(8)
Newton's Method
103(3)
Secant Method
106(2)
Remarks on Line Search Methods
108(5)
Exercises
109(4)
Gradient Methods
113(26)
Introduction
113(2)
The Method of Steepest Descent
115(7)
Analysis of Gradient Methods
122(17)
Convergence
122(7)
Convergence Rate
129(5)
Exercises
134(5)
Newton's Method
139(12)
Introduction
139(3)
Analysis of Newton's Method
142(3)
Levenberg-Marquardt Modification
145(1)
Newton's Method for Nonlinear Least-Squares
146(5)
Exercises
149(2)
Conjugate Direction Methods
151(16)
Introduction
151(2)
The Conjugate Direction Algorithm
153(5)
The Conjugate Gradient Algorithm
158(3)
The Conjugate Gradient Algorithm for Non-Quadratic Problems
161(6)
Exercises
164(3)
Quasi-Newton Methods
167(20)
Introduction
167(1)
Approximating the Inverse Hessian
168(3)
The Rank One Correction Formula
171(5)
The DFP Algorithm
176(4)
The BFGS Algorithm
180(7)
Exercises
184(3)
Solving Ax = b
187(32)
Least-Squares Analysis
187(9)
Recursive Least-Squares Algorithm
196(3)
Solution to Ax = b Minimizing ∥x∥
199(2)
Kaczmarz's Algorithm
201(3)
Solving Ax = b in General
204(15)
Exercises
212(7)
Unconstrained Optimization and Neural Networks
219(18)
Introduction
219(2)
Single-Neuron Training
221(3)
Backpropagation Algorithm
224(13)
Exercises
234(3)
Genetic Algorithms
237(18)
Basic Description
237(6)
Chromosomes and Representation Schemes
238(1)
Selection and Evolution
238(5)
Analysis of Genetic Algorithms
243(5)
Real-Number Genetic Algorithms
248(7)
Exercises
250(5)
Part III Linear Programming
Introduction to Linear Programming
255(32)
A Brief History of Linear Programming
255(2)
Simple Examples of Linear Programs
257(6)
Two-Dimensional Linear Programs
263(1)
Convex Polyhedra and Linear Programming
264(3)
Standard Form Linear Programs
267(5)
Basic Solutions
272(4)
Properties of Basic Solutions
276(3)
A Geometric View of Linear Programs
279(8)
Exercises
282(5)
Simplex Method
287(34)
Solving Linear Equations Using Row Operations
287(7)
The Canonical Augmented Matrix
294(1)
Updating the Augmented Matrix
295(2)
The Simplex Algorithm
297(6)
Matrix Form of the Simplex Method
303(4)
The Two-Phase Simplex Method
307(3)
The Revised Simplex Method
310(11)
Exercises
315(6)
Duality
321(18)
Dual Linear Programs
321(7)
Properties of Dual Problems
328(11)
Exercises
333(6)
Non-Simplex Methods
339(26)
Introduction
339(1)
Khachiyan's Method
340(3)
Affine Scaling Method
343(5)
Basic Algorithm
343(4)
Two-Phase Method
347(1)
Karmarkar's Method
348(17)
Basic Ideas
348(1)
Karmarkar's Canonical Form
349(2)
Karmarkar's Restricted Problem
351(1)
From General Form to Karmarkar's Canonical Form
352(4)
The Algorithm
356(4)
Exercises
360(5)
Part IV Nonlinear Constrained Optimization
Problems with Equality Constraints
365(32)
Introduction
365(1)
Problem Formulation
366(2)
Tangent and Normal Spaces
368(6)
Lagrange Condition
374(10)
Second-Order Conditions
384(3)
Minimizing Quadratics Subject to Linear Constraints
387(10)
Exercises
391(6)
Problems with Inequality Constraints
397(20)
Karush-Kuhn-Tucker Condition
397(9)
Second-Order Conditions
406(11)
Exercises
410(7)
Convex Optimization Problems
417(22)
Introduction
417(2)
Convex Functions
419(8)
Convex Optimization Problems
427(12)
Exercises
433(6)
Algorithms for Constrained Optimization
439(16)
Introduction
439(1)
Projections
439(2)
Projected Gradient Methods
441(4)
Penalty Methods
445(10)
Exercises
451(4)
References 455(7)
Index 462

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