IEEE Robotics & Automation Magazine - December 2013 - 164

The customizability and power of
FPGAs, as well as the ability to reuse
components in other designs, makes
them a powerful tool in the arsenal of
any roboticist. The prevalence and

rapid evolution of these devices in
industry make it easy to learn more
about FPGAs. This book aids in the
process, providing a clear introduction,
numerous examples, and an excellent

flow-a balance of technical detail and
informative discussion.
-Reviewed by
Alex Simpkins, Jr. , Ph.D.
San Diego, California

Convex Optimization
Steven Boyd and Lieven Vandenberghe,
Cambridge, United Kingdom, 2014,
716 pages.
onvex Optimization by Steven
Boyd and Lieven Vandenberghe is a classic text on optimization. Though there have
been significant recent developments in
the field of optimization, this text still
covers the fundamental concepts that
any researcher or practitioner needs to
know and certainly takes the reader far
beyond the basics. As robotic systems
gain in complexity, dimensionality, and
dynamic task challenges, convex optimization is playing an increasing role in all
aspects. It is useful in control systems,
circuit and mechanical design, statistics,
data analysis and modeling, and more.
As such, it is clear that roboticists should
study and incorporate convex optimization as part of their knowledge base.
The book is broken down into three
main sections ("Theory," "Application,"
and "Algorithms"), 11 chapters, and
three appendices. It also has a list of
references, an index, and a notation
list. Each chapter ends with a bibliography and exercises, making it easy to
take the chapter further and explore
beyond the book.
Chapter 1 gives a mathematical
overview of optimization, introducing
least squares and linear programming,
convex and nonlinear optimization,
local and global problems, and the role
of convex optimization in nonconvex
problems. Finally, the rest of the book
is outlined, and notation to be used
is described.

The "Theory" section is broken down
into Chapter 2 on convex sets, Chapter 3
on convex functions, Chapter 4 on convex optimization problems, and Chapter 5 on duality. Chapter 2 is a key
chapter, beginning with affine sets and
moving to convexity, then covering several relevant examples, convexity in
operations, and (perhaps most importantly) separating and supporting hyper
planes and dual cones. Chapter 3 discusses how convexity and quasiconvexity
as concepts can be extended to functions,
and functions of convexity (such as logconvex functions). The authors describe
that for many nonconvex functions, a
convex hull can be formed using
inequalities. Next, Chapter 4 covers convex optimization problems, standard
forms, equivalent problems, parameters,
local and global criteria, and more. Linear fractional programming is also
touched on, a topic particularly relevant
to uncertain and complex systems. Of
course, quadratic problems are also presented. Chapter 5 shows how convex
optimization problems that are difficult
to solve in their original formulation can
be transformed into Lagrangian dual
problems that can be solved. A good
book provides not only the what and
how but also the why, and here the
authors spend time motivating the dual
problem, constraints, sensitivity, examples, and criteria for feasibility.
The "Applications" section is broken
down into Chapter 6 on approximation
and fitting, Chapter 7 on statistical estimation, and Chapter 8 on geometric
problems. One of the most common
uses of optimization is for approximately (or exactly) solving equations.
The authors craft Chapter 6 into many
sections by first presenting the prob-

lem, then giving several interpretations
or applications. For example, the first
section begins with norm approximation problems, then presents approximation, estimation, geometric, and
design interpretations. Challenging
data and functions can be fit using convex optimization, and the authors highlight this well. Caveats and strategies for
successful application are discussed.
Statistical analysis is a challenging field
and often is not taught from the perspective of optimization. Chapter 7 discusses a variety of statistical problems
and how they can be framed then
solved within the convex optimization
framework. Maximum likelihood estimation is a fundamental concept when
dealing with parametric distributions,
and it is one of many concepts laid out
quite clearly here. This chapter, which
covers testing hypotheses and making
decisions, is useful since robots operating in uncertain environments need
these capabilities. When acting in the
world, geometric problems abound.
Chapter 8 addresses a variety of geometric applications that will be of great
interest to roboticists. For example,
problems dealing with classification
and polynomial discrimination are presented, along with solutions.
The third section of the book,
"Algorithms," contains Chapter 9 on
unconstrained optimization, Chapter
10 on equality constrained optimization, and Chapter 11 on interior-point
methods. Unconstrained minimization
problems, summarized in Chapter 9,
are difficult to solve analytically, so
typically a numerical approach is
taken. The authors here present not
only various methods and examples
but also tests for determining how

Digital Object Identifier 10.1109/MRA.2013.2283189
Date of publication: 6 December 2013

164

IEEE ROBOTICS & AUTOMATION MAGAZINE

DECEMBER 2013

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - December 2013