Sum of Squares: Theory and Applications PDF
Preview Sum of Squares: Theory and Applications
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. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. (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
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