An Unconditional Lower Bound for the Active-Set Method in Almost-Linear Programming

Abstract

We prove that, for every fixed $\epsilon \in (0,1]$, the active-set method requires exponentially many iterations in the worst case to maximize a convex objective of the form $(x_1{)}^{1+\epsilon }+{\vec{c}}^{\top }\vec{x}$ subject to linear constraints, regardless of the pivot rule used. Our lower bound therefore holds for objectives arbitrarily close to the linear case $\epsilon =0$, where the active-set method coincides with the simplex method and a lower bound for all pivot rules would constitute a major breakthrough. The special case $\epsilon =1$ yields a lower bound in convex quadratic maximization. This improves on the best previously known lower bound [IPCO 2025] requiring polynomial objectives of degree $\omega (\log d)$ in dimension $d$ and firmly resolves the open question of whether a constant degree suffices. Our result is based on a novel extended formulation, recursively constructed using deformed products. Its key feature is that it projects onto a polygonal approximation of a parabola while preserving all of its exponentially many vertices. We define a convex objective with a single non-linear variable that forces the active-set method to follow the parabolic boundary of this projection, without allowing any shortcuts along chords corresponding to edges of its full-dimensional preimage.