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A Unified Worst Case for Classical Simplex and Policy Iteration Pivot Rules

Authors Yann Disser , Nils Mosis

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Author Details

Yann Disser
  • TU Darmstadt, Germany
Nils Mosis
  • TU Darmstadt, Germany

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Yann Disser and Nils Mosis. A Unified Worst Case for Classical Simplex and Policy Iteration Pivot Rules. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.27

Abstract

We construct a family of Markov decision processes for which the policy iteration algorithm needs an exponential number of improving switches with Dantzig’s rule, with Bland’s rule, and with the Largest Increase pivot rule. This immediately translates to a family of linear programs for which the simplex algorithm needs an exponential number of pivot steps with the same three pivot rules. Our results yield a unified construction that simultaneously reproduces well-known lower bounds for these classical pivot rules, and we are able to infer that any (deterministic or randomized) combination of them cannot avoid an exponential worst-case behavior. Regarding the policy iteration algorithm, pivot rules typically switch multiple edges simultaneously and our lower bound for Dantzig’s rule and the Largest Increase rule, which perform only single switches, seem novel. Regarding the simplex algorithm, the individual lower bounds were previously obtained separately via deformed hypercube constructions. In contrast to previous bounds for the simplex algorithm via Markov decision processes, our rigorous analysis is reasonably concise.

Subject Classification

ACM Subject Classification
  • Theory of computation → Linear programming
  • Mathematics of computing → Markov processes
Keywords
  • Bland’s pivot rule
  • Dantzig’s pivot rule
  • Largest Increase pivot rule
  • Markov decision process
  • policy iteration
  • simplex algorithm

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