Self-Adapting Network Relaxations for Weakly Coupled Markov Decision Processes

High-dimensional weakly coupled Markov decision processes (WDPs) arise in dynamic decision making and reinforcement learning, decomposing into smaller Markov decision processes (MDPs) when linking constraints are relaxed. The Lagrangian relaxation of WDPs (LAG) exploits this property to compute policies and (optimistic) bounds efficiently; however, dualizing linking constraints averages away combinatorial information. We introduce feasibility network relaxations (FNRs), a new class of linear programming relaxations that exactly represents the linking constraints. We develop a procedure to obtain the unique minimally sized relaxation, which we refer to as self-adapting FNR, as its size automatically adjusts to the structure of the linking constraints. Our analysis informs model selection: (i) the self-adapting FNR provides (weakly) stronger bounds than LAG, is polynomially sized when linking constraints admit a tractable network representation, and can even be smaller than LAG, and (ii) self-adapting FNR provides bounds and policies that match the approximate linear programming (ALP) approach but is substantially smaller in size than the ALP formulation and a recent alternative Lagrangian that is equivalent to ALP. We perform numerical experiments on constrained dynamic assortment and preemptive maintenance applications. Our results show that self-adapting FNR significantly improves upon LAG in terms of policy performance and/or bounds, while being an order of magnitude faster than an alternative Lagrangian and ALP, which are unsolvable in several instances.