An unconditional lower bound for the active-set method on the hypercube
Journal:
arXiv
Published Date:
Feb 25, 2025
Abstract
The existence of a polynomial-time pivot rule for the simplex method is a
fundamental open question in optimization. While many super-polynomial lower
bounds exist for individual or very restricted classes of pivot rules, there
currently is little hope for an unconditional lower bound that addresses all
pivot rules. We approach this question by considering the active-set method as
a natural generalization of the simplex method to non-linear objectives. This
generalization allows us to prove the first unconditional lower bound for all
pivot rules. More precisely, we construct a multivariate polynomial of degree
linear in the number of dimensions such that the active-set method started in
the origin visits all vertices of the hypercube. We hope that our framework
serves as a starting point for a new angle of approach to understanding the
complexity of the simplex method.
