Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yann Disser, Nils Mosis

TL;DR

The paper addresses whether a universal, polynomial-time pivot rule exists for the simplex method by introducing the active-set method as a nonlinear generalization and proving the first unconditional lower bound for all pivot rules. It constructs, for each dimension $n$, a degree-$n$ multivariate polynomial $\mathcal{F}_{n}$ such that starting from $\boldsymbol{0}$ the active-set method requires $2^n-1$ iterations to maximize over the hypercube $[0,1]^n$, visiting every vertex in a simplex-like edge traversal. It then shows that the unique improving edge at each non-optimal vertex yields a Hamiltonian path on the hypercube, and that the induced orientation is a decomposable unique sink orientation (USO) solvable in linear time, highlighting that the approach does not embed NP-hardness. The results provide an unconditional, universal lower bound framework for simplex-type methods and open avenues to reduce polynomial degree or extend the setting to other polytopes while preserving exponential behavior, thereby guiding the search for fundamental limits on pivot rules.

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.

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

TL;DR

The paper addresses whether a universal, polynomial-time pivot rule exists for the simplex method by introducing the active-set method as a nonlinear generalization and proving the first unconditional lower bound for all pivot rules. It constructs, for each dimension , a degree- multivariate polynomial such that starting from the active-set method requires iterations to maximize over the hypercube , visiting every vertex in a simplex-like edge traversal. It then shows that the unique improving edge at each non-optimal vertex yields a Hamiltonian path on the hypercube, and that the induced orientation is a decomposable unique sink orientation (USO) solvable in linear time, highlighting that the approach does not embed NP-hardness. The results provide an unconditional, universal lower bound framework for simplex-type methods and open avenues to reduce polynomial degree or extend the setting to other polytopes while preserving exponential behavior, thereby guiding the search for fundamental limits on pivot rules.

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.

Paper Structure

This paper contains 11 sections, 11 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Our results.
  3. Notation.
  4. Simplex and active-set method
  5. Unconditional super-polynomial bounds for active-set
  6. Polynomials of linear degree
  7. Uniqueness of improving edges
  8. Visiting all vertices of the hypercube
  9. Proofs of the super-polynomial bounds
  10. The complexity of maximizing over the hypercube
  11. Future research

Key Result

theorem 1

For all $n\in\mathbb{N}_{>2}$, there is a multivariate polynomial $\mathcal{F}_{n}$ of degree $n$ such that the active-set method started in $\boldsymbol{0}$ needs $2^n-1$ iterations to optimize $\mathcal{F}_{n}$ over the $n$-dimensional hypercube $[0,1]^n$, irrespective of the pivot rule.

Theorems & Definitions (20)

  • theorem 1
  • corollary 1
  • proposition 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • proposition 2
  • definition 1
  • lemma 3
  • ...and 10 more