An Accelerated Fitted Value Iteration Algorithm for MDPs with Finite and Vector-Valued Action Space

Abstract

This paper studies an accelerated fitted value iteration (FVI) algorithm to solve high-dimensional Markov decision processes (MDPs). FVI is an approximate dynamic programming algorithm that has desirable theoretical properties. However, it can be intractable when the action space is finite but vector-valued. To solve such MDPs via FVI, we first approximate the value functions by a two-layer neural network (NN) with rectified linear units (ReLU) being activation functions. We then verify that such approximators are strong enough for the MDP. To speed up the FVI, we recast the action selection problem as a two-stage stochastic programming problem, where the resulting recourse function comes from the two-layer NN. Then, the action selection problem is solved with a specialized multi-cut decomposition algorithm. More specifically, we design valid cuts by exploiting the structure of the approximated value functions to update the actions. We prove that the decomposition can find the global optimal solution in a finite number of iterations and the overall accelerated FVI is consistent. Finally, we verify the performance of the FVI algorithm via a multi-facility capacity investment problem (MCIP). A comprehensive numerical study is implemented, where the results show that the FVI is significantly accelerated without sacrificing too much in precision.