Globally Optimizing Owing to Tensor Decomposition

Abstract

While global optimization is a challenging topic in the nonconvex setting, a recent approach for optimizing polynomials reformulates the problem as an equivalent problem on measures, which is called a moment problem. It is then relaxed into a convex semidefinite programming problem whose solution gives the first moments of a measure supporting the optimal points. However, extracting the global solutions to the polynomial problem from those moments is still difficult, especially if the latter are poorly estimated. In this paper, we address the issue of extracting optimal points and interpret it as a tensor decomposition problem. By leveraging tools developed for noisy tensor decomposition, we propose a method to find the global solutions to a polynomial optimization problem from a noisy estimation of the solution of its corresponding moment problem. Finally, the interest of tensor decomposition methods for global polynomial optimization is shown through a detailed case study.