Sum of Squares: Theory and Applications

154 Pages·2020·6.15 MB·English

by  Pablo A. Parrilo, Rekha R. Thomas

Proceedings of Symposia in
A M
PPLIED ATHEMATICS
Volume 77
Sum of Squares: Theory
and Applications
AMS Short Course
Sum of Squares: Theory and Applications
January 14–15, 2019
Baltimore, Maryland
Pablo A. Parrilo
Rekha R. Thomas
Editors
Volume 77
Sum of Squares: Theory
and Applications
AMS Short Course
Sum of Squares: Theory and Applications
January 14–15, 2019
Baltimore, Maryland
Pablo A. Parrilo
Rekha R. Thomas
Editors

Proceedings of Symposia in
A M
PPLIED ATHEMATICS
Volume 77
Sum of Squares: Theory
and Applications
AMS Short Course
Sum of Squares: Theory and Applications
January 14–15, 2019
Baltimore, Maryland
Pablo A. Parrilo
Rekha R. Thomas
Editors
2010 Mathematics Subject Classification. Primary05-XX,14-XX,52-XX,68-XX,90-XX.
For additional informationand updates on this book, visit
www.ams.org/bookpages/psapm-77
Library of Congress Cataloging-in-Publication Data
LibraryofCongressCataloging-in-PublicationData
Names: AmericanMathematicalSociety. ShortCourse,SumofSquares: TheoryandApplications
(2019 : Baltimore, Maryland), author. — Parrilo, Pablo A., editor. — Thomas, Rekha R.,
1967-editor.
Title: Sum of squares : theory and applications : AMS Short Course, Sum of Squares : Theory
and Applications, January 14–15, 2019, Baltimore, Maryland / Pablo A. Parrilo, Rekha R.
Thomas,editors.
Description: Providence, Rhode Island : American Mathematical Society, [2020] | Series: Pro-
ceedingsofsymposiainappliedmathematics,0160-7634;volume77|Includesbibliographical
referencesandindex.
Identifiers: LCCN2020012201|ISBN9781470450250(softcover)|ISBN9781470460402(ebook)
Subjects: LCSH: Polynomials. | Convex geometry. | Convex sets. | Geometry, Algebraic. |
Semidefinite programming. | Mathematical optimization. | AMS: Combinatorics. — Alge-
braic geometry. | Convex and discrete geometry. | Computer science. | Operations research,
mathematicalprogramming.
Classification: LCCQA161.P59A642020|DDC511/.6–dc23
LCrecordavailableathttps://lccn.loc.gov/2020012201
DOI:https://doi.org/10.1090/psapm/077/00677
Copying and reprinting. Individual readersofthispublication,andnonprofit librariesacting
for them, are permitted to make fair use of the material, such as to copy select pages for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
ispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requestsforpermission
toreuseportionsofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. For
moreinformation,pleasevisitwww.ams.org/publications/pubpermissions.
Sendreq[e-mail].
(cid:2)c 2020bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttps://www.ams.org/
10987654321 252423222120
Contents
Preface vii
A Brief Introduction to Sums of Squares
Grigoriy Blekherman 1
The Geometry of Spectrahedra
Cynthia Vinzant 11
Lifts of Convex Sets
Hamza Fawzi 37
Algebraic Geometry and Sums of Squares
Mauricio Velasco 59
Sums of Squares in Theoretical Computer Science
Ankur Moitra 83
Applications of Sums of Squares
Georgina Hall 103
v

Preface
This book is a compilation of the lecture notes from the 2019 AMS Short
Course on the theory and applications of sum of squares polynomials. Over the
past two decades, the theory of nonnegative and sums of squares polynomials has
becomerelevanttoseveralareasofmathematicsandrelatedfields. Theaimofthis
bookistoshowcase someoftherecentdevelopments inthefield. The introductory
chapter gives a brief overview of sums of squares and their connections to real
algebraic geometry and polynomial optimization, and also places all chapters in
context. The remaining five chapters explore different aspects of sums of squares.
The discussion begins with the geometry of spectrahedra, which are the feasible
regions of semidefinite programs. This ties in with recent results on expressing
convex sets as projections of spectrahedra, a topic further explored in the chapter
on lifts of convex sets. The next chapter explains recent generalizations, using
classical methods in algebraic geometry, of Hilbert’s 1888 theorem cataloging the
situations under which all nonnegative polynomials are sums of squares. This is
followed by a chapter on recent advances in theoretical computer science that have
been obtained by viewing sums of squares as a meta-algorithm for many standard
computationaltasks. Sincesumsofsquareshavefoundapplicationsinawidearray
ofpracticalareas,thebookendswithasurveyofseveralexamplesfromengineering,
statistics, and operations research.
Pablo A. Parrilo
Rekha R. Thomas
vii

ProceedingsofSymposiainAppliedMathematics
Volume77,2020
https://doi.org/10.1090/psapm/077/00680
A brief introduction to sums of squares
Grigoriy Blekherman
Abstract. We give a briefoverview andhistory ofnonnegative polynomials
andsumsofsquares. Thischapteralsoprovidesaguidetohowthesubsequent
chaptersconnecttothiscentraltheory.
Thischapterisaninformalintroductiontothetheoryofnonnegativepolynomi-
als and sums of squares. We assume that the reader is familiar with linear algebra,
and especially with positive semidefinite symmetric matrices. For more details on
these we recommend [6, Appendix A] and [3, Section II.12]. The study of nonneg-
ativity and its relation with sums of squares is a classical topic in real algebraic
geometry, starting with the work of Hilbert, who classified the cases in terms of
degree and number of variables where nonnegative polynomials are always sums
of squares of polynomials (Theorem 3.2). Generalizations of this theorem are the
subject of Mauricio Velasco’s chapter. Hilbert’s 17th problem asked whether any
globally nonnegative polynomial is a sum of squares of rational functions. Artin’s
positive solution of Hilbert’s 17th problem in the 1920’s, using the Artin-Schreier
theory of real closed fields left a lasting imprint on the subject.
The study of the subject changed substantially in the early 2000’s when it was
realized by Lasserre and Parrilo, following earlier work by N.Z. Shor and Choi,
Lam and Reznick, that one can use semidefinite programming to efficiently search
for sums of squares certificates of nonnegativity in practice [9,14,19,25]. The
resulting hierarchies (called Lasserre, moment or sums of squares hierarchies) were
motivated by theorems of Schmu¨dgen and Putinar on representations of nonneg-
ative polynomials as sums of squares on compact semialgebraic sets (see Section
5). The sum of squares method has many applications in engineering, robotics and
computer science. For more details see Georgina Hall’s chapter.
1. Two guiding questions
Howcanwedecideifafunction,forinstanceapolynomialinonevariable,takes
only nonnegative values? Consider for example, p(x)=5x4−4x3−x2+2x+2. It
is not immediately clear, although quite elementary to show, that p(x) is globally
nonnegative. However, if we write p(x) as a sum of squares of polynomials:
p(x)=(x2+1)2+(2x2−x−1)2,
then its nonnegativity is readily apparent.
2010 MathematicsSubjectClassification. Primary14P05,14P10,90C22.
(cid:2)c2020 American Mathematical Society
1