Majorizing a Multivariate Polynomial Over the Unit Sphere
Jan de Leeuw · eScholarship (California Digital Library) · 2011
MAJORIZING A MULTIVARIATE POLYNOMIAL OVER THE UNIT SPHERE JAN DE LEEUW 1. P ROBLEM The problem studied in this note is to minimize a polynomial P : R m ⇒ R over the unit sphere S = {x | x x = 1}. Clearly the problem is well-defined, because the minimum always exists. We use the standard notation P(x) = ∑ α p α x α for multivariate polynomials, where α are vectors of m integers, and m x α = ∏ x j j . j=1 One important application we have in mind is minimizing polynomial func- tions of Jacobi plane rotations [De Leeuw, 2008], a problem that occurs in factor analysis, component analysis, multiway decomposition, and multi- dimensional scaling. Instead of using sin(θ ) and cos(θ ) which define the Jacobi rotation, we parametrize using x 1 and x 2 satisfying x 1 + x 2 = 1. This means using a bivariate polynomial on the sphere instead of a univariate trigoniometric polynomial. Date: Friday 17 th June, 2011 — 8h 52min — Typeset in T IMES R OMAN . 2000 Mathematics Subject Classification. 49M20. Key words and phrases. Majorization, Polynomial Optimization.