Low rank decompositions for sum of squares optimization

Abstract

In this thesis, we investigate theoretical and numerical advantages of a novel representation for Sum of Squares (SOS) decomposition of univariate and multivariate polynomials. This representation formulates a SOS problem by interpolating a polynomial at a finite set of sampling points. As compared to the conventional coefficient method of SOS, the formulation has a low rank property in its constraints. The low rank property is desirable as it improves computation speed for calculations of barrier gradient and Hessian assembling in many semidefinite programming (SDP) solvers. Currently, SDPT3 solver has a function to store low rank constraints to explore its numerical advantages. Some SOS examples are constructed and tested on SDPT3 to a great extent. The experimental results demonstrate that the computation time decreases significantly. Moreover, the solutions of the interpolation method are verified to be numerically more stable and accurate than the solutions yielded from the coefficient method.