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Practical Optimization - Algorithms and Engineering Applications

by

Andreas Antoniou
Wu-Sheng Lu
Department of Electrical and Computer Engineering
University of Victoria, Canada

2007 Springer Science+Business Media, LLC

Preface

The rapid advancements in the efficiency of digital computers and the evolution
of reliable software for numerical computation during the past three
decades have led to an astonishing growth in the theory, methods, and algorithms
of numerical optimization. This body of knowledge has, in turn, motivated
widespread applications of optimization methods in many disciplines,
e.g., engineering, business, and science, and led to problem solutions that were
considered intractable not too long ago.
Although excellent books are available that treat the subject of optimization
with great mathematical rigor and precision, there appears to be a need for a
book that provides a practical treatment of the subject aimed at a broader audience
ranging from college students to scientists and industry professionals.
This book has been written to address this need. It treats unconstrained and
constrained optimization in a unified manner and places special attention on the
algorithmic aspects of optimization to enable readers to apply the various algorithms
and methods to specific problems of interest. To facilitate this process,
the book provides many solved examples that illustrate the principles involved,
and includes, in addition, two chapters that deal exclusively with applications of
unconstrained and constrained optimization methods to problems in the areas of
pattern recognition, control systems, robotics, communication systems, and the
design of digital filters. For each application, enough background information
is provided to promote the understanding of the optimization algorithms used
to obtain the desired solutions.
Chapter 1 gives a brief introduction to optimization and the general structure
of optimization algorithms. Chapters 2 to 9 are concerned with unconstrained
optimization methods. The basic principles of interest are introduced in Chapter
2. These include the first-order and second-order necessary conditions for
a point to be a local minimizer, the second-order sufficient conditions, and the
optimization of convex functions. Chapter 3 deals with general properties of
algorithms such as the concepts of descent function, global convergence, and
XVI
rate of convergence. Chapter 4 presents several methods for one-dimensional
optimization, which are commonly referred to as line searches. The chapter
also deals with inexact line-search methods that have been found to increase
the efficiency in many optimization algorithms. Chapter 5 presents several
basic gradient methods that include the steepest descent, Newton, and Gauss-
Newton methods. Chapter 6 presents a class of methods based on the concept of
conjugate directions such as the conjugate-gradient, Fletcher-Reeves, Powell,
and Partan methods. An important class of unconstrained optimization methods
known as quasi-Newton methods is presented in Chapter 7. Representative
methods of this class such as the Davidon-Fletcher-Powell and Broydon-
Fletcher-Goldfarb-Shanno methods and their properties are investigated. The
chapter also includes a practical, efficient, and reliable quasi-Newton algorithm
that eliminates some problems associated with the basic quasi-Newton method.
Chapter 8 presents minimax methods that are used in many applications including
the design of digital filters. Chapter 9 presents three case studies in
which several of the unconstrained optimization methods described in Chapters
4 to 8 are applied to point pattern matching, inverse kinematics for robotic
manipulators, and the design of digital filters.
Chapters 10 to 16 are concerned with constrained optimization methods.
Chapter 10 introduces the fundamentals of constrained optimization. The concept
of Lagrange multipliers, the first-order necessary conditions known as
Karush-Kuhn-Tucker conditions, and the duality principle of convex programming
are addressed in detail and are illustrated by many examples. Chapters
11 and 12 are concerned with linear programming (LP) problems. The general
properties of LP and the simplex method for standard LP problems are
addressed in Chapter 11. Several interior-point methods including the primal
affine-scaling, primal Newton-barrier, and primal dual-path following methods
are presented in Chapter 12. Chapter 13 deals with quadratic and general
convex programming. The so-called active-set methods and several interiorpoint
methods for convex quadratic programming are investigated. The chapter
also includes the so-called cutting plane and ellipsoid algorithms for general
convex programming problems. Chapter 14 presents two special classes of convex
programming known as semidefinite and second-order cone programming,
which have found interesting applications in a variety of disciplines. Chapter
15 treats general constrained optimization problems that do not belong to the
class of convex programming; special emphasis is placed on several sequential
quadratic programming methods that are enhanced through the use of efficient
line searches and approximations of the Hessian matrix involved. Chapter 16,
which concludes the book, examines several applications of constrained optimization
for the design of digital filters, for the control of dynamic systems, for
evaluating the force distribution in robotic systems, and in multiuser detection
for wireless communication systems.
PREFACE xvii
The book also includes two appendices, A and B, which provide additional
support material. Appendix A deals in some detail with the relevant parts of
linear algebra to consolidate the understanding of the underlying mathematical
principles involved whereas Appendix B provides a concise treatment of the
basics of digital filters to enhance the understanding of the design algorithms
included in Chaps. 8, 9, and 16.
The book can be used as a text for a sequence of two one-semester courses
on optimization. The first course comprising Chaps. 1 to 7, 9, and part of
Chap. 10 may be offered to senior undergraduate or first-year graduate students.
The prerequisite knowledge is an undergraduate mathematics background of
calculus and linear algebra. The material in Chaps. 8 and 10 to 16 may be
used as a text for an advanced graduate course on minimax and constrained
optimization. The prerequisite knowledge for thi^ course is the contents of the
first optimization course.
The book is supported by online solutions of the end-of-chapter problems
under password as well as by a collection of MATLAB programs for free access
by the readers of the book, which can be used to solve a variety of optimization
problems. These materials can be downloaded from the book's website:
http://www.ece.uvic.ca/~optimization/.
We are grateful to many of our past students at the University of Victoria,
in particular, Drs. M. L. R. de Campos, S. Netto, S. Nokleby, D. Peters, and
Mr. J. Wong who took our optimization courses and have helped improve the
manuscript in one way or another; to Chi-Tang Catherine Chang for typesetting
the first draft of the manuscript and for producing most of the illustrations; to
R. Nongpiur for checking a large part of the index; and to R Ramachandran
for proofreading the entire manuscript. We would also like to thank Professors
M. Ahmadi, C. Charalambous, P. S. R. Diniz, Z. Dong, T. Hinamoto, and P. P.
Vaidyanathan for useful discussions on optimization theory and practice; Tony
Antoniou of Psicraft Studios for designing the book cover; the Natural Sciences
and Engineering Research Council of Canada for supporting the research that
led to some of the new results described in Chapters 8, 9, and 16; and last but
not least the University of Victoria for supporting the writing of this book over
anumber of years.
Andreas Antoniou and Wu-Sheng Lu

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