New Dependencies of Hierarchies in Polynomial Optimization
Adam Kurpisz, Timo de Wolff · 2019
We compare four key hierarchies for solving Constrained Polynomial Optimization Problems (CPOP) arising from semialgebraic proof systems: Sum of Squares (SOS), Sum of Diagonally Dominant Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams (SA) hierarchies. We prove a collection of dependencies among these hierarchies both for general CPOPs and for optimization problems on the Boolean hypercube. Key results include for the general case that the SONC and SOS hierarchy are polynomially incomparable, while SDSOS is contained in SONC. On the Boolean hypercube, we show as a main result that Schmudgen-like versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent. Moreover, we show that SA* is contained in any Schmudgen-like hierarchy that provides a O(n) degree bound.