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An introduction to optimization / Edwin K.P. Chong and Stanislaw H. çZak.

By: Chong, Edwin Kah Pin.
Contributor(s): Zak, Stanislaw H.
Material type: materialTypeLabelBookSeries: Wiley-Interscience series in discrete mathematics and optimization.Publisher: New York : Wiley, 2001Edition: 2nd ed.Description: xv, 476 p. : ill. ; 25 cm. + hbk.ISBN: 0471391263 (hbk.).Subject(s): Mathematical optimizationDDC classification: 519.3
Contents:
Part I: Mathematical review -- Methods of proof and some notation -- Vector spaces and matrices -- Transformations -- Concepts from geometry -- Elements of calculus -- Part II: Unconstrained optimization -- Basics of set-constrained and unconstrained optimization -- One-dimensional search methods -- Gradient methods -- Newton's method -- Conjugate direction methods -- Quasi-Newton methods -- Solving Ax=b -- Unconstrained optimization and neural networks -- Genetic algorithms -- Part III: Linear programming -- Introduction to linear programming -- Simplex method -- Duality -- Non-simplex methods -- Part IV: Nonlinear constrained optimization -- Problems with equality constraints -- Problems with inequality constraints -- Convex optimization problems -- Algorithms for constrained optimization.

Enhanced descriptions from Syndetics:

A modern, up-to-date introduction to optimization theory and methods
This authoritative book serves as an introductory text to optimization at the senior undergraduate and beginning graduate levels. With consistently accessible and elementary treatment of all topics, An Introduction to Optimization, Second Edition helps students build a solid working knowledge of the field, including unconstrained optimization, linear programming, and constrained optimization.
Supplemented with more than one hundred tables and illustrations, an extensive bibliography, and numerous worked examples to illustrate both theory and algorithms, this book also provides:
* A review of the required mathematical background material
* A mathematical discussion at a level accessible to MBA and business students
* A treatment of both linear and nonlinear programming
* An introduction to recent developments, including neural networks, genetic algorithms, and interior-point methods
* A chapter on the use of descent algorithms for the training of feedforward neural networks
* Exercise problems after every chapter, many new to this edition
* MATLAB(r) exercises and examples
* Accompanying Instructor's Solutions Manual available on request
An Introduction to Optimization, Second Edition helps students prepare for the advanced topics and technological developments that lie ahead. It is also a useful book for researchers and professionals in mathematics, electrical engineering, economics, statistics, and business.

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

A Wiley-Interscience publication.

Includes bibliographical references (pages 455-461) and index.

Part I: Mathematical review -- Methods of proof and some notation -- Vector spaces and matrices -- Transformations -- Concepts from geometry -- Elements of calculus -- Part II: Unconstrained optimization -- Basics of set-constrained and unconstrained optimization -- One-dimensional search methods -- Gradient methods -- Newton's method -- Conjugate direction methods -- Quasi-Newton methods -- Solving Ax=b -- Unconstrained optimization and neural networks -- Genetic algorithms -- Part III: Linear programming -- Introduction to linear programming -- Simplex method -- Duality -- Non-simplex methods -- Part IV: Nonlinear constrained optimization -- Problems with equality constraints -- Problems with inequality constraints -- Convex optimization problems -- Algorithms for constrained optimization.

Table of contents provided by Syndetics

  • Preface (p. xiii)
  • Part I Mathematical Review
  • 1 Methods of Proof and Some Notation (p. 1)
  • 1.1 Methods of Proof (p. 1)
  • 1.2 Notation (p. 3)
  • Exercises (p. 4)
  • 2 Vector Spaces and Matrices (p. 5)
  • 2.1 Real Vector Spaces (p. 5)
  • 2.2 Rank of a Matrix (p. 10)
  • 2.3 Linear Equations (p. 14)
  • 2.4 Inner Products and Norms (p. 16)
  • Exercises (p. 19)
  • 3 Transformations (p. 21)
  • 3.1 Linear Transformations (p. 21)
  • 3.2 Eigenvalues and Eigenvectors (p. 22)
  • 3.3 Orthogonal Projections (p. 25)
  • 3.4 Quadratic Forms (p. 26)
  • 3.5 Matrix Norms (p. 31)
  • Exercises (p. 35)
  • 4 Concepts from Geometry (p. 39)
  • 4.1 Line Segments (p. 39)
  • 4.2 Hyperplanes and Linear Varieties (p. 39)
  • 4.3 Convex Sets (p. 42)
  • 4.4 Neighborhoods (p. 44)
  • 4.5 Polytopes and Polyhedra (p. 45)
  • Exercises (p. 47)
  • 5 Elements of Calculus (p. 49)
  • 5.1 Sequences and Limits (p. 49)
  • 5.2 Differentiability (p. 55)
  • 5.3 The Derivative Matrix (p. 57)
  • 5.4 Differentiation Rules (p. 59)
  • 5.5 Level Sets and Gradients (p. 60)
  • 5.6 Taylor Series (p. 64)
  • Exercises (p. 68)
  • Part II Unconstrained Optimization
  • 6 Basics of Set-Constrained and Unconstrained Optimization (p. 73)
  • 6.1 Introduction (p. 73)
  • 6.2 Conditions for Local Minimizers (p. 75)
  • Exercises (p. 83)
  • 7 One-Dimensional Search Methods (p. 91)
  • 7.1 Golden Section Search (p. 91)
  • 7.2 Fibonacci Search (p. 95)
  • 7.3 Newton's Method (p. 103)
  • 7.4 Secant Method (p. 106)
  • 7.5 Remarks on Line Search Methods (p. 108)
  • Exercises (p. 109)
  • 8 Gradient Methods (p. 113)
  • 8.1 Introduction (p. 113)
  • 8.2 The Method of Steepest Descent (p. 115)
  • 8.3 Analysis of Gradient Methods (p. 122)
  • 8.3.1 Convergence (p. 122)
  • 8.3.2 Convergence Rate (p. 129)
  • Exercises (p. 134)
  • 9 Newton's Method (p. 139)
  • 9.1 Introduction (p. 139)
  • 9.2 Analysis of Newton's Method (p. 142)
  • 9.3 Levenberg-Marquardt Modification (p. 145)
  • 9.4 Newton's Method for Nonlinear Least-Squares (p. 146)
  • Exercises (p. 149)
  • 10 Conjugate Direction Methods (p. 151)
  • 10.1 Introduction (p. 151)
  • 10.2 The Conjugate Direction Algorithm (p. 153)
  • 10.3 The Conjugate Gradient Algorithm (p. 158)
  • 10.4 The Conjugate Gradient Algorithm for Non-Quadratic Problems (p. 161)
  • Exercises (p. 164)
  • 11 Quasi-Newton Methods (p. 167)
  • 11.1 Introduction (p. 167)
  • 11.2 Approximating the Inverse Hessian (p. 168)
  • 11.3 The Rank One Correction Formula (p. 171)
  • 11.4 The DFP Algorithm (p. 176)
  • 11.5 The BFGS Algorithm (p. 180)
  • Exercises (p. 184)
  • 12 Solving Ax = b (p. 187)
  • 12.1 Least-Squares Analysis (p. 187)
  • 12.2 Recursive Least-Squares Algorithm (p. 196)
  • 12.3 Solution to Ax = b Minimizing [double vertical line]x[double vertical line] (p. 199)
  • 12.4 Kaczmarz's Algorithm (p. 201)
  • 12.5 Solving Ax = b in General (p. 204)
  • Exercises (p. 212)
  • 13 Unconstrained Optimization and Neural Networks (p. 219)
  • 13.1 Introduction (p. 219)
  • 13.2 Single-Neuron Training (p. 221)
  • 13.3 Backpropagation Algorithm (p. 224)
  • Exercises (p. 234)
  • 14 Genetic Algorithms (p. 237)
  • 14.1 Basic Description (p. 237)
  • 14.1.1 Chromosomes and Representation Schemes (p. 238)
  • 14.1.2 Selection and Evolution (p. 238)
  • 14.2 Analysis of Genetic Algorithms (p. 243)
  • 14.3 Real-Number Genetic Algorithms (p. 248)
  • Exercises (p. 250)
  • Part III Linear Programming
  • 15 Introduction to Linear Programming (p. 255)
  • 15.1 A Brief History of Linear Programming (p. 255)
  • 15.2 Simple Examples of Linear Programs (p. 257)
  • 15.3 Two-Dimensional Linear Programs (p. 263)
  • 15.4 Convex Polyhedra and Linear Programming (p. 264)
  • 15.5 Standard Form Linear Programs (p. 267)
  • 15.6 Basic Solutions (p. 272)
  • 15.7 Properties of Basic Solutions (p. 276)
  • 15.8 A Geometric View of Linear Programs (p. 279)
  • Exercises (p. 282)
  • 16 Simplex Method (p. 287)
  • 16.1 Solving Linear Equations Using Row Operations (p. 287)
  • 16.2 The Canonical Augmented Matrix (p. 294)
  • 16.3 Updating the Augmented Matrix (p. 295)
  • 16.4 The Simplex Algorithm (p. 297)
  • 16.5 Matrix Form of the Simplex Method (p. 303)
  • 16.6 The Two-Phase Simplex Method (p. 307)
  • 16.7 The Revised Simplex Method (p. 310)
  • Exercises (p. 315)
  • 17 Duality (p. 321)
  • 17.1 Dual Linear Programs (p. 321)
  • 17.2 Properties of Dual Problems (p. 328)
  • Exercises (p. 333)
  • 18 Non-Simplex Methods (p. 339)
  • 18.1 Introduction (p. 339)
  • 18.2 Khachiyan's Method (p. 340)
  • 18.3 Affine Scaling Method (p. 343)
  • 18.3.1 Basic Algorithm (p. 343)
  • 18.3.2 Two-Phase Method (p. 347)
  • 18.4 Karmarkar's Method (p. 348)
  • 18.4.1 Basic Ideas (p. 348)
  • 18.4.2 Karmarkar's Canonical Form (p. 349)
  • 18.4.3 Karmarkar's Restricted Problem (p. 351)
  • 18.4.4 From General Form to Karmarkar's Canonical Form (p. 352)
  • 18.4.5 The Algorithm (p. 356)
  • Exercises (p. 360)
  • Part IV Nonlinear Constrained Optimization
  • 19 Problems with Equality Constraints (p. 365)
  • 19.1 Introduction (p. 365)
  • 19.2 Problem Formulation (p. 366)
  • 19.3 Tangent and Normal Spaces (p. 368)
  • 19.4 Lagrange Condition (p. 374)
  • 19.5 Second-Order Conditions (p. 384)
  • 19.6 Minimizing Quadratics Subject to Linear Constraints (p. 387)
  • Exercises (p. 391)
  • 20 Problems with Inequality Constraints (p. 397)
  • 20.1 Karush-Kuhn-Tucker condition (p. 397)
  • 20.2 Second-Order Conditions (p. 406)
  • Exercises (p. 410)
  • 21 Convex Optimization Problems (p. 417)
  • 21.1 Introduction (p. 417)
  • 21.2 Convex Functions (p. 419)
  • 21.3 Convex Optimization Problems (p. 427)
  • Exercises (p. 433)
  • 22 Algorithms for Constrained Optimization (p. 439)
  • 22.1 Introduction (p. 439)
  • 22.2 Projections (p. 439)
  • 22.3 Projected Gradient Methods (p. 441)
  • 22.4 Penalty Methods (p. 445)
  • Exercises (p. 451)
  • References (p. 455)
  • Index (p. 462)

Author notes provided by Syndetics

Edwin K. P. Chong, PhD, is Professor of Electrical and Computer Engineering at Colorado State University, Fort Collins, Colorado
Stanislaw H. Zak, PhD, is Professor in the School of Electrical and Computer Engineering at Purdue University, West Lafayette, Indiana