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An Introduction to Optimization, 4th Edition

Book Description

Praise for the Third Edition ". . . guides and leads the reader through the learning path . . . [e]xamples are stated very clearly and the results are presented with attention to detail." —MAA Reviews

Fully updated to reflect new developments in the field, the Fourth Edition of Introduction to Optimization fills the need for accessible treatment of optimization theory and methods with an emphasis on engineering design. Basic definitions and notations are provided in addition to the related fundamental background for linear algebra, geometry, and calculus.

This new edition explores the essential topics of unconstrained optimization problems, linear programming problems, and nonlinear constrained optimization. The authors also present an optimization perspective on global search methods and include discussions on genetic algorithms, particle swarm optimization, and the simulated annealing algorithm. Featuring an elementary introduction to artificial neural networks, convex optimization, and multi-objective optimization, the Fourth Edition also offers:

  • A new chapter on integer programming

  • Expanded coverage of one-dimensional methods

  • Updated and expanded sections on linear matrix inequalities

  • Numerous new exercises at the end of each chapter

  • MATLAB exercises and drill problems to reinforce the discussed theory and algorithms

  • Numerous diagrams and figures that complement the written presentation of key concepts

  • MATLAB M-files for implementation of the discussed theory and algorithms (available via the book's website)

  • Introduction to Optimization, Fourth Edition is an ideal textbook for courses on optimization theory and methods. In addition, the book is a useful reference for professionals in mathematics, operations research, electrical engineering, economics, statistics, and business.

    Table of Contents

    1. Cover
    2. Half Title page
    3. Title page
    4. Copyright page
    5. Dedication
    6. Preface
    7. Part I: Mathematical Review
      1. Chapter 1: Methods of Proof and Some Notation
        1. 1.1 Methods of Proof
        2. 1.2 Notation
        3. Exercises
      2. Chapter 2: Vector Spaces and Matrices
        1. 2.1 Vector and Matrix
        2. 2.2 Rank of a Matrix
        3. 2.3 Linear Equations
        4. 2.4 Inner Products and Norms
        5. Exercises
      3. Chapter 3: Transformations
        1. 3.1 Linear Transformations
        2. 3.2 Eigenvalues and Eigenvectors
        3. 3.3 Orthogonal Projections
        4. 3.4 Quadratic Forms
        5. 3.5 Matrix Norms
        6. Exercises
      4. Chapter 4: Concepts from Geometry
        1. 4.1 Line Segments
        2. 4.2 Hyperplanes and Linear Varieties
        3. 4.3 Convex Sets
        4. 4.4 Neighborhoods
        5. 4.5 Polytopes and Polyhedra
        6. Exercises
      5. Chapter 5: Elements of Calculus
        1. 5.1 Sequences and Limits
        2. 5.2 Differentiability
        3. 5.3 The Derivative Matrix
        4. 5.4 Differentiation Rules
        5. 5.5 Level Sets and Gradients
        6. 5.6 Taylor Series
        7. Exercises
    8. Part II: Unconstrained Optimization
      1. Chapter 6: Basics of Set-Constrained and Unconstrained Optimization
        1. 6.1 Introduction
        2. 6.2 Conditions for Local Minimizers
        3. Exercises
      2. Chapter 7: One-Dimensional Search Methods
        1. 7.1 Introduction
        2. 7.2 Golden Section Search
        3. 7.3 Fibonacci Method
        4. 7.4 Bisection Method
        5. 7.5 Newton’s Method
        6. 7.6 Secant Method
        7. 7.7 Bracketing
        8. 7.8 Line Search in Multidimensional Optimization
        9. Exercises
      3. Chapter 8: Gradient Methods
        1. 8.1 Introduction
        2. 8.2 The Method of Steepest Descent
        3. 8.3 Analysis of Gradient Methods
        4. Exercises
      4. Chapter 9: Newton’s Method
        1. 9.1 Introduction
        2. 9.2 Analysis of Newton’s Method
        3. 9.3 Levenberg-Marquardt Modification
        4. 9.4 Newton’s Method for Nonlinear Least Squares
        5. Exercises
      5. Chapter 10: Conjugate Direction Methods
        1. 10.1 Introduction
        2. 10.2 The Conjugate Direction Algorithm
        3. 10.3 The Conjugate Gradient Algorithm
        4. 10.4 The Conjugate Gradient Algorithm for Nonquadratic Problems
        5. Exercises
      6. Chapter 11: Quasi-Newton Methods
        1. 11.1 Introduction
        2. 11.2 Approximating the Inverse Hessian
        3. 11.3 The Rank One Correction Formula
        4. 11.4 The DFP Algorithm
        5. 11.5 The BFGS Algorithm
        6. Exercises
      7. Chapter 12: Solving Linear Equations
        1. 12.1 Least-Squares Analysis
        2. 12.2 The Recursive Least-Squares Algorithm
        3. 12.3 Solution to a Linear Equation with Minimum Norm
        4. 12.4 Kaczmarz’s Algorithm
        5. 12.5 Solving Linear Equations in General
        6. Exercises
      8. Chapter 13: Unconstrained Optimization and Neural Networks
        1. 13.1 Introduction
        2. 13.2 Single-Neuron Training
        3. 13.3 The Backpropagation Algorithm
        4. Exercises
      9. Chapter 14: Global Search Algorithms
        1. 14.1 Introduction
        2. 14.2 The Nelder-Mead Simplex Algorithm
        3. 14.3 Simulated Annealing
        4. 14.4 Particle Swarm Optimization
        5. 14.5 Genetic Algorithms
        6. Exercises
    9. Part III: Linear Programming
      1. Chapter 15: Introduction to Linear Programming
        1. 15.1 Brief History of Linear Programming
        2. 15.2 Simple Examples of Linear Programs
        3. 15.3 Two-Dimensional Linear Programs
        4. 15.4 Convex Polyhedra and Linear Programming
        5. 15.5 Standard Form Linear Programs
        6. 15.6 Basic Solutions
        7. 15.7 Properties of Basic Solutions
        8. 15.8 Geometric View of Linear Programs
        9. Exercises
      2. Chapter 16: Simplex Method
        1. 16.1 Solving Linear Equations Using Row Operations
        2. 16.2 The Canonical Augmented Matrix
        3. 16.3 Updating the Augmented Matrix
        4. 16.4 The Simplex Algorithm
        5. 16.5 Matrix Form of the Simplex Method
        6. 16.6 Two-Phase Simplex Method
        7. 16.7 Revised Simplex Method
        8. Exercises
      3. Chapter 17: Duality
        1. 17.1 Dual Linear Programs
        2. 17.2 Properties of Dual Problems
        3. Exercises
      4. Chapter 18: Nonsimplex Methods
        1. 18.1 Introduction
        2. 18.2 Khachiyan’s Method
        3. 18.3 Affine Scaling Method
        4. 18.4 Karmarkar’s Method
        5. Exercises
      5. Chapter 19: Integer Linear Programming
        1. 19.1 Introduction
        2. 19.2 Unimodular Matrices
        3. 19.3 The Gomory Cutting-Plane Method
        4. Exercises
    10. Part IV: Nonlinear Constrained Optimization
      1. Chapter 20: Problems with Equality Constraints
        1. 20.1 Introduction
        2. 20.2 Problem Formulation
        3. 20.3 Tangent and Normal Spaces
        4. 20.4 Lagrange Condition
        5. 20.5 Second-Order Conditions
        6. 20.6 Minimizing Quadratics Subject to Linear Constraints
        7. Exercises
      2. Chapter 21: Problems with Inequality Constraints
        1. 21.1 Karush-Kuhn-Tucker Condition
        2. 21.2 Second-Order Conditions
        3. Exercises
      3. Chapter 22: Convex Optimization Problems
        1. 22.1 Introduction
        2. 22.2 Convex Functions
        3. 22.3 Convex Optimization Problems
        4. 22.4 Semidefinite Programming
        5. Exercises
      4. Chapter 23: Algorithms for Constrained Optimization
        1. 23.1 Introduction
        2. 23.2 Projections
        3. 23.3 Projected Gradient Methods with Linear Constraints
        4. 23.4 Lagrangian Algorithms
        5. 23.5 Penalty Methods
        6. Exercises
      5. Chapter 24: Multiobjective Optimization
        1. 24.1 Introduction
        2. 24.2 Pareto Solutions
        3. 24.3 Computing the Pareto Front
        4. 24.4 From Multiobjective to Single-Objective Optimization
        5. 24.5 Uncertain Linear Programming Problems
        6. Exercises
    11. References
    12. Index