Rethinking Optimal Pivoting Paths of Simplex Method

Abstract

The optimal pivoting path of the simplex method is crucial for solving linear programming problems. This study proposes a new method to provide the optimal pivot and all the corresponding best pivoting paths using the Monte Carlo tree search (MCTS). First, we propose SimplexPseudoTree to transfer the simplex method into tree search mode while avoiding repeated basis variables. Based on the SimplexPseudoTree, we propose four reinforcement learning (RL) models with two actions and two rewards to make the Monte Carlo tree search suitable for the simplex method. In addition, we set a new action selection criterion to ameliorate the inaccurate evaluation in the initial exploration. We experimentally validated that the proposed schedule can avoid unnecessary searches and provide the optimal pivot. Additionally, we theoretically proved that our method determines the minimum number of pivot iterations and all corresponding optimal pivoting paths. Furthermore, this method provides the best labels for the pivot rules based on supervised learning.