Mean-variance optimization in Markov decision processes

We consider finite horizon Markov decision processes under performance measures that involve both the mean and the variance of the cumulative reward. We show that either randomized or history-based policies can im-prove performance. We prove that the com-plexity of computing a policy that maximizes the mean reward under a variance constraint is NP-hard for some cases, and strongly NP-hard for others. We finally offer pseudopoly-nomial exact and approximation algorithms. 1