-

 

 

 




Optimization Online





 

Polynomial Norms

Amir Ali Ahmadi(a_a_a***at***princeton.edu)
Etienne de Klerk(e.deklerk***at***uvt.nl)
Georgina Hall(gh4***at***princeton.edu)

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 testing feasibility of a semidefinite program (of possibly large size). 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-sos-convexity and extend a result of Reznick on sum of squares representation of positive definite forms to positive definite biforms. We conclude with some applications of polynomial norms to statistics and dynamical systems.

Keywords: polynomial norms, sum of squares polynomials, convex polynomials, semidefinite programming

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Nonlinear Optimization

Category 3: Convex and Nonsmooth Optimization

Citation:

Download: [PDF]

Entry Submitted: 05/15/2017
Entry Accepted: 05/15/2017
Entry Last Modified: 05/15/2017

Modify/Update this entry


  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository

 

Submit
Update
Policies
Coordinator's Board
Classification Scheme
Credits
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society