New Dependencies of Hierarchies in Polynomial Optimization

We compare four key hierarchies for solving Constrained Polynomial Optimization Problems (CPOP): 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. A direct consequence is the non-existence of a Putinar-like Positivstellensatz for SDSOS. On the Boolean hypercube, we show as a main result that Schm\"udgen-like versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent. Moreover, we show that SA* is contained in any Schm\"udgen-like hierarchy that provides a O(n) degree bound.Comment: 26 pages, 4 figure

Stable Set Polytopes with High Lift-and-Project Ranks for the Lov\'asz-Schrijver SDP Operator

We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lov{\'a}sz--Schrijver SDP operator $\text{LS}_+$, with a particular focus on a search for relatively small graphs with high $\text{LS}_+$-rank (the least number of iterations of the $\text{LS}_+$ operator on the fractional stable set polytope to compute the stable set polytope). We provide families of graphs whose $\text{LS}_+$-rank is asymptotically a linear function of its number of vertices, which is the best possible up to improvements in the constant factor (previous best result in this direction, from 1999, yielded graphs whose $\text{LS}_+$-rank only grew with the square root of the number of vertices). We also provide several new $\text{LS}_+$-minimal graphs, most notably a $12$-vertex graph with $\text{LS}_+$-rank $4$, and study the properties of a vertex-stretching operation that appears to be promising in generating $\text{LS}_+$-minimal graphs

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th