Finite-State Approximations to Discounted and Average Cost Constrained Markov Decision Processes

TL;DR

This paper develops methods for approximating constrained Markov decision processes with finite states, proving convergence of the optimal values and providing explicit bounds on the approximation error for both discounted and average cost criteria.

Contribution

It introduces a unified approach for finite-state approximation of constrained MDPs, establishing convergence and error bounds under regularity conditions.

Findings

Optimal value convergence proven for finite-state models

Explicit error bounds for approximation quality

Method for computing approximately optimal policies

Abstract

In this paper, we consider the finite-state approximation of a discrete-time constrained Markov decision process (MDP) under the discounted and average cost criteria. Using the linear programming formulation of the constrained discounted cost problem, we prove the asymptotic convergence of the optimal value of the finite-state model to the optimal value of the original model. With further continuity condition on the transition probability, we also establish a method to compute approximately optimal policies. For the average cost, instead of using the finite-state linear programming approximation method, we use the original problem definition to establish the finite-state asymptotic approximation of the constrained problem and compute approximately optimal policies. Under Lipschitz type regularity conditions on the components of the MDP, we also obtain explicit rate of convergence bounds quantifying how the approximation improves as the size of the approximating finite state space increases.