Concave quadratic lower bound on a general polytope

Construct a concave quadratic polynomial over some polytope for which the active-set method requires super-polynomially many iterations, irrespective of the pivot rule.

Background

The paper emphasizes that, for concave quadratic objectives, only weakly polynomial algorithms such as the ellipsoid and interior-point methods are known, while the active-set method is a candidate for a strongly polynomial algorithm because of its combinatorial structure. A super-polynomial lower bound applying to all pivot rules in this setting would clarify the limitations of that approach and would extend the paper’s unconditional lower-bound framework beyond its constructed higher-degree objectives.