  This paper introduces algorithms for problems where a decision maker has to
control a system composed of several components and has access to only partial
information on the state of each component. Such problems are difficult because
of the partial observations, and because of the curse of dimensionality that
appears when the number of components increases. Partially observable Markov
decision processes (POMDPs) have been introduced to deal with the first
challenge, while weakly coupled stochastic dynamic programs address the second.
Drawing from these two branches of the literature, we introduce the notion of
weakly coupled POMDPs. The objective is to find a policy maximizing the total
expected reward over a finite horizon. Our algorithms rely on two ingredients.
The first, which can be used independently, is a mixed integer linear
formulation for generic POMDPs that computes an optimal memoryless policy. The
formulation is strengthened with valid cuts based on a probabilistic
interpretation of the dependence between random variables, and its linear
relaxation provide a practically tight upper bound on the value of an optimal
history-dependent policies. The second is a collection of mathematical
programming formulations and algorithms which provide tractable policies and
upper bounds for weakly coupled POMDPs. Lagrangian relaxations, fluid
approximations, and almost sure constraints relaxations enable to break the
curse of dimensionality. We test our generic POMDPs formulations on benchmark
instance forms the literature, and our weakly coupled POMDP algorithms on a
maintenance problem. Numerical experiments show the efficiency of our approach.
