We study the min max optimization problem introduced in [Fonteneau et al. (2011), ``Towards min max reinforcement learning'', Springer CCIS, vol. 129, pp. 61-77] for computing policies for batch mode reinforcement learning in a deterministic setting with fixed, finite time horizon. First, we show that the min part of this problem is NP-hard. We then provide two relaxation schemes. The first relaxation scheme works by dropping some constraints in order to obtain a problem that is solvable in polynomial time. The second relaxation scheme, based on a Lagrangian relaxation where all constraints are dualized, can also be solved in polynomial time. We also theoretically prove and empirically illustrate that both relaxation schemes provide better results than those given in [Fonteneau et al. (2011)
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