1st Edition

A First Course in Optimization

By Charles Byrne Copyright 2015
    316 Pages
    by Chapman & Hall

    Give Your Students the Proper Groundwork for Future Studies in Optimization

    A First Course in Optimization is designed for a one-semester course in optimization taken by advanced undergraduate and beginning graduate students in the mathematical sciences and engineering. It teaches students the basics of continuous optimization and helps them better understand the mathematics from previous courses.

    The book focuses on general problems and the underlying theory. It introduces all the necessary mathematical tools and results. The text covers the fundamental problems of constrained and unconstrained optimization as well as linear and convex programming. It also presents basic iterative solution algorithms (such as gradient methods and the Newton–Raphson algorithm and its variants) and more general iterative optimization methods.

    This text builds the foundation to understand continuous optimization. It prepares students to study advanced topics found in the author’s companion book, Iterative Optimization in Inverse Problems, including sequential unconstrained iterative optimization methods.

    Optimization without Calculus
    Chapter Summary
    The Arithmetic Mean-Geometric Mean Inequality
    An Application of the AGM Inequality: the Number e
    Extending the AGM Inequality
    Optimization Using the AGM Inequality
    The Holder and Minkowski Inequalities
    Cauchy's Inequality
    Optimizing using Cauchy's Inequality
    An Inner Product for Square Matrices
    Discrete Allocation Problems

    Geometric Programming
    Chapter Summary
    An Example of a GP Problem
    Posynomials and the GP Problem
    The Dual GP Problem
    Solving the GP Problem
    Solving the DGP Problem
    Constrained Geometric Programming

    Basic Analysis
    Chapter Summary
    Minima and Infima
    Limsup and Liminf
    Another View

    Convex Sets
    Chapter Summary
    The Geometry of Real Euclidean Space
    A Bit of Topology
    Convex Sets in RJ
    More on Projections
    Linear and Affine Operators on RJ
    The Fundamental Theorems
    Block-Matrix Notation
    Theorems of the Alternative
    Another Proof of Farkas' Lemma
    Gordan's Theorem Revisited

    Vector Spaces and Matrices
    Chapter Summary
    Vector Spaces
    Basic Linear Algebra
    LU and QR Factorization
    The LU Factorization

    Linear Programming
    Chapter Summary
    Primal and Dual
    Converting a Problem to PS Form
    Duality Theorems
    The Basic Strong Duality Theorem
    Another Proof
    Proof of Gale's Strong Duality Theorem
    Some Examples
    The Simplex Method
    Yet Another Proof
    The Sherman–Morrison–Woodbury Identity
    An Example of the Simplex Method
    Another Example
    Some Possible Difficulties
    Topics for Projects

    Matrix Games and Optimization
    Chapter Summary
    Two-Person Zero-Sum Games
    Deterministic Solutions
    Randomized Solutions
    Symmetric Games
    Positive Games
    Example: The "Bluffing" Game
    Learning the Game
    Non-Constant-Sum Games

    Chapter Summary
    Directional Derivative
    Partial Derivatives
    Some Examples
    Gâteaux Derivative
    Fréchet Derivative
    The Chain Rule

    Convex Functions
    Chapter Summary
    Functions of a Single Real Variable
    Functions of Several Real Variables
    Sub-Differentials and Sub-Gradients
    Sub-Gradients and Directional Derivatives
    Functions and Operators
    Convex Sets and Convex Functions

    Convex Programming
    Chapter Summary
    The Primal Problem
    From Constrained to Unconstrained
    Saddle Points
    The Karush–Kuhn–Tucker Theorem
    On Existence of Lagrange Multipliers
    The Problem of Equality Constraints
    Two Examples
    The Dual Problem
    Nonnegative Least-Squares Solutions
    An Example in Image Reconstruction
    Solving the Dual Problem
    Minimum One-Norm Solutions

    Iterative Optimization
    Chapter Summary
    The Need for Iterative Methods
    Optimizing Functions of a Single Real Variable
    The Newton–Raphson Approach
    Approximate Newton–Raphson Methods
    Derivative-Free Methods
    Rates of Convergence
    Descent Methods
    Optimizing Functions of Several Real Variables
    Auxiliary-Function Methods
    Projected Gradient-Descent Methods
    Feasible-Point Methods
    Quadratic Programming
    Simulated Annealing

    Solving Systems of Linear Equations
    Chapter Summary
    Arbitrary Systems of Linear Equations
    Nonnegative Systems of Linear Equations
    Regularized SMART and EMML
    Block-Iterative Methods

    Conjugate-Direction Methods
    Chapter Summary
    Iterative Minimization
    Quadratic Optimization
    Conjugate Bases for RJ
    The Conjugate Gradient Method
    Krylov Subspaces
    Extensions of the CGM

    Chapter Summary
    Contraction Operators
    Orthogonal-Projection Operators
    Two Useful Identities
    Averaged Operators
    Gradient Operators
    Affine-Linear Operators
    Paracontractive Operators
    Matrix Norms

    Looking Ahead
    Chapter Summary
    Sequential Unconstrained Minimization
    Examples of SUM
    Auxiliary-Function Methods
    The SUMMA Class of AF Methods



    Exercises appear at the end of each chapter.


    Charles Byrne

    "This book, split in fifteen chapters, presents a thorough introduction to optimization for undergraduate and graduate courses in science and engineering and forms a useful toolkit of optimization techniques across several domains. … Throughout this very interesting book, a number of solved and unsolved exercises provide the reader with all the necessary tools to understand the techniques and their applicability in real-life problems."
    Zentralblatt MATH 1312