1st Edition

Numerical Methods and Optimization An Introduction

By Sergiy Butenko, Panos M. Pardalos Copyright 2014
    414 Pages 53 B/W Illustrations
    by Chapman & Hall

    For students in industrial and systems engineering (ISE) and operations research (OR) to understand optimization at an advanced level, they must first grasp the analysis of algorithms, computational complexity, and other concepts and modern developments in numerical methods. Satisfying this prerequisite, Numerical Methods and Optimization: An Introduction combines the materials from introductory numerical methods and introductory optimization courses into a single text. This classroom-tested approach enriches a standard numerical methods syllabus with optional chapters on numerical optimization and provides a valuable numerical methods background for students taking an introductory OR or optimization course.

    The first part of the text introduces the necessary mathematical background, the digital representation of numbers, and different types of errors associated with numerical methods. The second part explains how to solve typical problems using numerical methods. Focusing on optimization methods, the final part presents basic theory and algorithms for linear and nonlinear optimization.

    The book assumes minimal prior knowledge of the topics. Taking a rigorous yet accessible approach to the material, it includes some mathematical proofs as samples of rigorous analysis but in most cases, uses only examples to illustrate the concepts. While the authors provide a MATLAB® guide and code available for download, the book can be used with other software packages.

    Sets and Functions
    Fundamental Theorem of Algebra
    Vectors and Linear (Vector) Spaces
    Matrices and Their Properties
    Preliminaries from Real and Functional Analysis

    Numbers and Errors
    Conversion between Different Number Systems
    Floating Point Representation of Numbers
    Definitions of Errors
    Round-off Errors

    Numerical Methods for Standard Problems
    Elements of Numerical Linear Algebra
    Direct Methods for Solving Systems of Linear Equations
    Iterative Methods for Solving Systems of Linear Equations
    Overdetermined Systems and Least Squares Solution
    Stability of a Problem
    Computing Eigenvalues and Eigenvectors

    Solving Equations
    Fixed Point Method
    Bracketing Methods
    Newton’s Method
    Secant Method
    Solution of Nonlinear Systems

    Polynomial Interpolation
    Forms of Polynomials
    Polynomial Interpolation Methods
    Theoretical Error of Interpolation and Chebyshev Polynomials

    Numerical Integration
    Trapezoidal Rule
    Simpson's Rule
    Precision and Error of Approximation
    Composite Rules
    Using Integrals to Approximate Sums

    Numerical Solution of Differential Equations
    Solution of a Differential Equation
    Taylor Series and Picard’s Methods
    Euler's Method
    Runge-Kutta Methods
    Systems of Differential Equations
    Higher-Order Differential Equations

    Introduction to Optimization
    Basic Concepts
    Formulating an Optimization Problem
    Mathematical Description
    Local and Global Optimality
    Existence of an Optimal Solution
    Level Sets and Gradients
    Convex Sets, Functions, and Problems

    Complexity Issues
    Algorithms and Complexity
    Average Running Time
    Randomized Algorithms
    Basics of Computational Complexity Theory
    Complexity of Local Optimization
    Optimal Methods for Nonlinear Optimization

    Introduction to Linear Programming
    Formulating a Linear Programming Model
    Examples of LP Models
    Practical Implications of Using LP Models
    Solving Two-Variable LPs Graphically
    Classification of LPs

    The Simplex Method for Linear Programming
    The Standard Form of LP
    The Simplex Method
    Geometry of the Simplex Method
    The Simplex Method for a General LP
    The Fundamental Theorem of LP
    The Revised Simplex Method
    Complexity of the Simplex Method

    Duality and Sensitivity Analysis in Linear Programming
    Defining the Dual LP
    Weak Duality and the Duality Theorem
    Extracting an Optimal Solution of the Dual LP from an Optimal Tableau of the Primal LP
    Correspondence between the Primal and Dual LP Types
    Complementary Slackness
    Economic Interpretation of the Dual LP
    Sensitivity Analysis

    Unconstrained Optimization
    Optimality Conditions
    Optimization Problems with a Single Variable
    Algorithmic Strategies for Unconstrained Optimization
    Method of Steepest Descent
    Newton’s Method
    Conjugate Direction Method
    Quasi-Newton Methods
    Inexact Line Search

    Constrained Optimization
    Optimality Conditions
    Projected Gradient Methods
    Sequential Unconstrained Minimization

    Notes and References




    Sergiy Butenko, Panos M. Pardalos

    "The book is in most parts very well developed and is served by nice illustrations, a fluid style of writing, and a layout that makes it easy to read. … [it will] serve well its purpose of bridging the gap between numerical analysis, operations research, and mathematical optimization for undergraduate students in the applied sciences."
    Mathematical Reviews, August 2014

    "If you are looking for an enjoyable and useful introduction to the basic topics of numerical methods and optimization, this is the right text to read. The authors are not only experienced lecturers but also active researchers in this area. They present the basic topics of numerical methods and optimization in an easy-to-follow, yet rigorous manner. In particular, they gently introduce some important topics, such as computational complexity, which are usually unavailable in textbooks on optimization for engineers. The authors occasionally turn to mathematical humor (such as ‘There are 10 types of people—those who understand binary, and those who don't’) to illustrate some material in the text. This informal contact with the reader exemplifies the engaging style of exposition characteristic of this excellent book."
    —Oleg Burdakov, Linkoeping University, Sweden