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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 ReviewsFully 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 programmingExpanded coverage of one-dimensional methodsUpdated and expanded sections on linear matrix inequalitiesNumerous new exercises at the end of each chapterMATLAB exercises and drill problems to reinforce the discussed theory and algorithmsNumerous diagrams and figures that complement the written presentation of key conceptsMATLAB 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.

An Introduction to Optimization

Preface xiiiPART I MATHEMATICAL REVIEW1 Methods of Proof and Some Notation 31.1 Methods of Proof 31.2 Notation 5Exercises 62 Vector Spaces and Matrices 72.1 Vector and Matrix 72.2 Rank of a Matrix 132.3 Linear Equations 172.4 Inner Products and Norms 19Exercises 223 Transformations 253.1 Linear Transformations 253.2 Eigenvalues and Eigenvectors 263.3 Orthogonal Projections 293.4 Quadratic Forms 313.5 Matrix Norms 35Exercises 404 Concepts from Geometry 454.1 Line Segments 454.2 Hyperplanes and Linear Varieties 464.3 Convex Sets 484.4 Neighborhoods 504.5 Polytopes and Polyhedra 52Exercises 535 Elements of Calculus 555.1 Sequences and Limits 555.2 Differentiability 625.3 The Derivative Matrix 635.4 Differentiation Rules 675.5 Level Sets and Gradients 685.6 Taylor Series 72Exercises 77PART II UNCONSTRAINED OPTIMIZATION6 Basics of Set-Constrained and Unconstrained Optimization 816.1 Introduction 816.2 Conditions for Local Minimizers 83Exercises 937 One-Dimensional Search Methods 1037.1 Introduction 1037.2 Golden Section Search 1047.3 Fibonacci Method 1087.4 Bisection Method 1167.5 Newton's Method 1167.6 Secant Method 1207.7 Bracketing 1237.8 Line Search in Multidimensional Optimization 124Exercises 1268 Gradient Methods 1318.1 Introduction 1318.2 The Method of Steepest Descent 1338.3 Analysis of Gradient Methods 141Exercises 1539 Newton's Method 1619.1 Introduction 1619.2 Analysis of Newton's Method 1649.3 Levenberg-Marquardt Modification 1689.4 Newton's Method for Nonlinear Least Squares 168Exercises 17110 Conjugate Direction Methods 17510.1 Introduction 17510.2 The Conjugate Direction Algorithm 17710.3 The Conjugate Gradient Algorithm 18210.4 The Conjugate Gradient Algorithm for NonquadraticProblems 186Exercises 18911 Quasi-Newton Methods 19311.1 Introduction 19311.2 Approximating the Inverse Hessian 19411.3 The Rank One Correction Formula 19711.4 The DFP Algorithm 20211.5 The BFGS Algorithm 207Exercises 21112 Solving Linear Equations 21712.1 Least-Squares Analysis 21712.2 The Recursive Least-Squares Algorithm 22712.3 Solution to a Linear Equation with Minimum Norm 23112.4 Kaczmarz's Algorithm 23212.5 Solving Linear Equations in General 236Exercises 24413 Unconstrained Optimization and Neural Networks 25313.1 Introduction 25313.2 Single-Neuron Training 25613.3 The Backpropagation Algorithm 258Exercises 27014 Global Search Algorithms 27314.1 Introduction 27314.2 The Nelder-Mead Simplex Algorithm 27414.3 Simulated Annealing 27814.4 Particle Swarm Optimization 28214.5 Genetic Algorithms 285Exercises 298PART III LINEAR PROGRAMMING15 Introduction to Linear Programming 30515.1 Brief History of Linear Programming 30515.2 Simple Examples of Linear Programs 30715.3 Two-Dimensional Linear Programs 31415.4 Convex Polyhedra and Linear Programming 31615.5 Standard Form Linear Programs 31815.6 Basic Solutions 32415.7 Properties of Basic Solutions 32715.8 Geometric View of Linear Programs 330Exercises 33516 Simplex Method 33916.1 Solving Linear Equations Using Row Operations 33916.2 The Canonical Augmented Matrix 34616.3 Updating the Augmented Matrix 34916.4 The Simplex Algorithm 35016.5 Matrix Form of the Simplex Method 35716.6 Two-Phase Simplex Method 36116.7 Revised Simplex Method 364Exercises 36917 Duality 37917.1 Dual Linear Programs 37917.2 Properties of Dual Problems 387Exercises 39418 Nonsimplex Methods 40318.1 Introduction 40318.2 Khachiyan's Method 40518.3 Affine Scaling Method 40818.4 Karmarkar's Method 413Exercises 42619 Integer Linear Programming 42919.1 Introduction 42919.2 Unimodular Matrices 43019.3 The Gomory Cutting-Plane Method 437Exercises 447PART IV NONLINEAR CONSTRAINED OPTIMIZATION20 Problems with Equality Constraints 45320.1 Introduction 45320.2 Problem Formulation 45520.3 Tangent and Normal Spaces 45620.4 Lagrange Condition 46320.5 Second-Order Conditions 47220.6 Minimizing Quadratics Subject to Linear Constraints 476Exercises 48121 Problems with Inequality Constraints 48721.1 Karush-Kuhn-Tucker Condition 48721.2 Second-Order Conditions 496Exercises 50122 Convex Optimization Problems 50922.1 Introduction 50922.2 Convex Functions 51222.3 Convex Optimization Problems 52122.4 Semidefinite Programming 527Exercises 54023 Algorithms for Constrained Optimization 54923.1 Introduction 54923.2 Projections 54923.3 Projected Gradient Methods with Linear Constraints 55323.4 Lagrangian Algorithms 55723.5 Penalty Methods 564Exercises 57124 Multiobjective Optimization 57724.1 Introduction 57724.2 Pareto Solutions 57824.3 Computing the Pareto Front 58124.4 From Multiobjective to Single-Objective Optimization 58524.5 Uncertain Linear Programming Problems 588Exercises 596References 599Index 609