@article{d1fbc4ed02944e1198c1093d423e186b,
title = "Polynomial norms",
abstract = " In this paper, we study polynomial norms, i.e., norms that are the dth root of a degree-d homogeneous polynomial f. We first show that a necessary and sufficient condition for f 1 /d to be a norm is for f to be strictly convex, or equivalently, convex and positive definite. Though not all norms come from dth roots of polynomials, we prove that any norm can be approximated arbitrarily well by a polynomial norm. We then investigate the computational problem of testing whether a form gives a polynomial norm. We show that this problem is strongly NP-hard already when the degree of the form is 4, but can always be answered by solving a hierarchy of semidefinite programs. We further study the problem of optimizing over the set of polynomial norms using semidefinite programming. To do this, we introduce the notion of r-sum of squares-convexity and extend a result of Reznick on sum of squares representations of positive definite forms to positive definite biforms. We conclude with some applications of polynomial norms to statistics and dynamical systems. ",
keywords = "Convex polynomials, Polynomial norms, Semidefinite programming, Sum of squares polynomials",
author = "Ahmadi, {Amir A.L.I.} and Klerk, {Etienne D.E.} and Georgina Hall",
note = "Funding Information: ∗Received by the editors February 26, 2018; accepted for publication (in revised form) August 23, 2018; published electronically February 5, 2019. http://www.siam.org/journals/siopt/29-1/M117284.html Funding: Amir Ali Ahmadi and Georgina Hall are partially supported by the DARPA Young Faculty Award, the Young Investigator Award of the AFOSR, the CAREER Award of the NSF, the Google Faculty Award, and the Sloan Fellowship. †ORFE, Princeton University, Princeton, NJ 08540 (a a a@princeton.edu). ‡Department Econometrics and Operations Research, TISEM, Tilburg University, 5000LE Tilburg, The Netherlands (e.deklerk@uvt.nl). §Corresponding author. Decision Sciences, INSEAD, Fontainebleau, France (georgina.hall@ insead.edu). Funding Information: Amir Ali Ahmadi and Georgina Hall are partially supported by the DARPA Young Faculty Award, the Young Investigator Award of the AFOSR, the CAREER Award of the NSF, the Google Faculty Award, and the Sloan Fellowship. The authors would like to thank an anonymous referee for suggesting the proof of the first part of Theorem 3.1, which improves our previous statement by quantifying the quality of the approximation as a function of n and d, and two other anonymous referees for constructive comments that considerably helped improve the draft. Publisher Copyright: {\textcopyright} 2019 Society for Industrial and Applied Mathematics.",
year = "2019",
doi = "10.1137/18M1172843",
language = "English (US)",
volume = "29",
pages = "399--422",
journal = "SIAM Journal on Optimization",
issn = "1052-6234",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "1",
}