  We consider lift-and-project methods for combinatorial optimization problems
and focus mostly on those lift-and-project methods which generate polyhedral
relaxations of the convex hull of integer solutions. We introduce many new
variants of Sherali--Adams and Bienstock--Zuckerberg operators. These new
operators fill the spectrum of polyhedral lift-and-project operators in a way
which makes all of them more transparent, easier to relate to each other, and
easier to analyze. We provide new techniques to analyze the worst-case
performances as well as relative strengths of these operators in a unified way.
In particular, using the new techniques and a result of Mathieu and Sinclair
from 2009, we prove that the polyhedral Bienstock--Zuckerberg operator requires
at least $\sqrt{2n}- \frac{3}{2}$ iterations to compute the matching polytope
of the $(2n+1)$-clique. We further prove that the operator requires
approximately $\frac{n}{2}$ iterations to reach the stable set polytope of the
$n$-clique, if we start with the fractional stable set polytope. Lastly, we
show that some of the worst-case instances for the positive semidefinite
Lov\'asz--Schrijver lift-and-project operator are also bad instances for the
strongest variants of the Sherali--Adams operator with positive semidefinite
strengthenings, and discuss some consequences for integrality gaps of convex
relaxations.
