Sparse SOS Relaxations for Minimizing Functions that are Summation of Small Polynomials

This paper discusses how to find the global minimum and minimizers of functions that are given as the summation of small polynomials (“small” means involving a small number of variables). Some sparse sum of squares (SOS) relaxations are proposed. We compare the computational complexity and lower bound with prior SOS relaxations. The proposed methods are specially useful in solving nonlinear least squares problems. Some numerical implementations of testing problems and randomly generated polynomials are given, which show that the proposed methods significantly improve the computational performance of prior work on exploiting sparsity. We also show the applications of this problem in solving polynomial systems derived from nonlinear differential equations and sensor network localization.