On the optimal pivot path of simplex method for linear programming based on reinforcement learning

Abstract

Based on the existing pivot rules, the simplex method for linear programming is not polynomial in the worst case. Therefore the optimal pivot of the simplex method is crucial. This study proposes the optimal rule to find all shortest pivot paths of the simplex method for linear programming problems based on Monte Carlo tree search (MCTS). Specifically, we first propose the SimplexPseudoTree to transfer the simplex method into tree search mode while avoiding repeated basis variables. Secondly, 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. Thirdly, we set a new action selection criterion to ameliorate the inaccurate evaluation in the initial exploration. It is proved that when the number of vertices in the feasible region is $C_n^m$, our method can generate all the shortest pivot paths, which is the polynomial of the number of variables. In addition, we experimentally validate that the proposed schedule can avoid unnecessary search and provide the optimal pivot path. Furthermore, this method can provide the best pivot labels for all kinds of supervised learning methods to solve linear programming problems.