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Amir Ali Ahmadi(a_a_aprinceton.edu) Abstract: In this paper, we study polynomial norms, i.e. norms that are the dth root of a degreed 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 NPhard 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 rsosconvexity 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 (Semidefinite Programming ) Category 2: Nonlinear Optimization Category 3: Convex and Nonsmooth Optimization Citation: Download: [PDF] Entry Submitted: 05/15/2017 Modify/Update this entry  
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