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This book provides the foundations of the theory of nonlinear optimization as well as some related algorithms and presents a variety of applications from diverse areas of applied sciences. The author combines three pillars of optimization—theoretical and algorithmic foundation, familiarity with various applications, and the ability to apply the theory and algorithms on actual problems—and rigorously and gradually builds the connection between theory, algorithms, applications, and implementation.
More than 170 theoretical, algorithmic, and numerical exercises that deepen and enhance the reader's understanding of the topics
several subjects not typically found in optimization books—for example, optimality conditions in sparsity-constrained optimization, hidden convexity, and total least squares
a large number of applications discussed theoretically and algorithmically, such as circle fitting, Chebyshev center, the Fermat–Weber problem, denoising, clustering, total least squares, and orthogonal regression and
theoretical and algorithmic topics demonstrated by the MatLAB toolbox CVX and a package of m-files that is posted on the book’s web site.
This book is intended for graduate or advanced undergraduate students of mathematics, computer science, and electrical engineering as well as other engineering departments. The book will also be of interest to researchers.
Mathematical Preliminaries
The Space n
The Space m×n
Inner Products and Norms
Eigenvalues and Eigenvectors
Basic Topological Concepts
Exercises
Optimality Conditions for Unconstrained Optimization
Global and Local Optima
Classification of Matrices
Second Order Optimality Conditions
Global Optimality Conditions
Quadratic Functions
Exercises
Least Squares
Solution of Overdetermined Systems
Data Fitting
Regularized Least Squares
Denoising
Nonlinear Least Squares
Circle Fitting
Exercises
The Gradient Method
Descent Directions Methods
The Gradient Method
The Condition Number
Diagonal Scaling
The Gauss–Newton Method
The Fermat–Weber Problem
Convergence Analysis of the Gradient Method
Exercises
Newton’s Method
Pure Newton’s Method
Damped Newton’s Method
The Cholesky Factorization
Exercises
Convex Sets
Definition and Examples
Algebraic Operations with Convex Sets
The Convex Hull
Convex Cones
Topological Properties of Convex Sets
Extreme Points
Exercises
Convex Functions
Definition and Examples
First Order Characterizations of Convex Functions
Second Order Characterization of Convex Functions
Operations Preserving Convexity
Level Sets of Convex Functions
Continuity and Differentiability of Convex Functions
Extended Real-Valued Functions
MAXIMA of Convex Functions
Convexity and Inequalities
Exercises
Convex Optimization
Definition
Examples
The Orthogonal Projection Operator
CVX
Exercises
Optimization over a Convex Set
Stationarity
Stationarity in Convex Problems
The Orthogonal Projection Revisited
The Gradient Projection Method
Sparsity Constrained Problems
Exercises
Optimality Conditions for Linearly Constrained Problems
Separation and Alternative Theorems
The KKT conditions
Orthogonal Regression
Exercises
The KKT Conditions
Inequality Constrained Problems
Inequality and Equality Constrained Problems
The Convex Case
Constrained Least Squares
Second Order Optimality Conditions
Optimality Conditions for the Trust Region Subproblem
Total Least Squares
Exercises
Duality
Motivation and Definition
Strong Duality in the Convex Case
Examples
Exercises
Bibliographic Notes