Constrained Markov decision processes with total cost criteria: Lagrangian approach and dual linear program

The aim of this paper is to investigate the Lagrangian approach and a related Linear Programming (LP) that appear in constrained Markov decision processes (CMDPs) with a countable state space and total expected cost criteria (of which the expected discounted cost is a special case). We consider transient MDPs and MDPs with uniform Lyapunov functions, and obtain for these an LP which is the dual of another one that has been shown to provide the optimal values and stationary policies. We show that there is no duality gap between these LPs under appropriate conditions. In obtaining the Linear Program for the general transient case, we establish, in particular, a calculation approach for the value function of the CMDP based on finite state approximation. Unlike previous approaches for state approximations for CMDPs (most of which were derived for the contracting framework), we do not need here any Slater type condition. We finally present another type of LP that allows the computation of optimal mixed stationary-deterministic policies.