Harmonic Hierarchies for Polynomial Optimization on Compact Gelfand Pairs

Abstract

The main topic of this thesis is a method of polynomial optimization on the unit sphere through hierarchies originating from harmonic analysis and representation theory. A graded optimization hierarchy for optimizing a polynomial $f(x)$ consists of a series of simpler problems of a given degree whose solutions approximate the optimum of $f(x)$ as the degree increases. We managed to prove that our optimization hierarchy has a the rate of convergence is $O(1/s^2)$. Furthermore, this thesis studies the foundations of a general framework to construct such optimization hierarchies in more general algebraic varieties. We studied the theory of harmonic analysis in Gelfand pairs and developed analogous techniques for a wider class of algebraic varieties, yielding similar but not as precise results as in the sphere.