This version
forms the basis for a forthcoming publication with Cambridge
University Press. Please
e-mail
me about any mistakes you spot or suspect, be it typos or
more serious things: I appreciate your input, and there may
still be time to make the change before it goes to print. For
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About the book
This is a book about optimization on smooth manifolds for
readers who are comfortable with linear algebra and
multivariable calculus. There are no prerequisites in geometry
or optimization. Chapters 3 and 5 in particular can serve as a
standalone introduction to differential and Riemannian geometry.
They focus on embedded submanifolds of linear spaces, with full
proofs. A distinguishing feature is that these early chapters
highlight computability and do not involve charts.
Chapter 8
provides the general theory so that we can build quotient
manifolds in Chapter 9. The optimization algorithms in Chapters
4 and 6 apply to the general case, but can already be understood
after reading Chapters 3 and 5. Chapter 7 details examples of
submanifolds that come up in practice. Chapter 10 covers more
advanced Riemannian tools, and Chapter 11 introduces geodesic
convexity.
In a
one-semester graduate course of the mathematics department at
Princeton University in 2019 and 2020 (24 lectures of 80 minutes
each, two projects, no exercises), I covered much of Chapters
1–6 and select parts of Chapter 7 before the midterm
break, then much of Chapters 8–9 and select parts of
Chapters 10–11 after the break. Those chapters were
shorter at the time, but it still made for a sustained pace. At
EPFL in 2021, I discussed mostly Chapters 1–8 in 13
lectures of 90 minutes, plus as many exercise sessions and two
projects. The course is popular with applied mathematicians and
mathematically inclined engineering students, at the graduate
and advanced undergraduate level.
How to cite this version
@Booklet{boumal2022intromanifolds,
title = {An
introduction to optimization on smooth manifolds},
author = {Boumal,
Nicolas},
howpublished = {To appear with Cambridge University
Press},
month =
{Apr},
year =
{2022},
url =
{http://www.nicolasboumal.net/book},
}
Page numbers will change a lot between this version and the
forthcoming published version. For now, it is best to reference
section numbers as these should be stable.