An unconditional lower bound for the active-set method on the hypercube

TL;DR

This paper establishes the first unconditional lower bound for all pivot rules by demonstrating that the active-set method can visit all vertices of a hypercube, highlighting fundamental complexity limitations.

Contribution

It introduces a polynomial-based framework to prove an unconditional lower bound for the active-set method, extending understanding of pivot rule complexity.

Findings

Active-set method can visit all hypercube vertices from the origin.

First unconditional lower bound for all pivot rules in optimization.

Framework may inspire new approaches to analyzing simplex method complexity.

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.