Mean-Variance Optimization and Algorithm for Finite-Horizon Markov Decision Processes

Multi-period mean-variance optimization is a long-standing problem, caused by the failure of dynamic programming principle. This paper studies the mean-variance optimization in a setting of finite-horizon discrete-time Markov decision processes (MDPs), where the objective is to maximize the combined metrics of mean and variance of the accumulated rewards at terminal stage. By introducing the concepts of pseudo mean and pseudo variance, we convert the original mean-variance MDP to a bilevel MDP, where the outer is a single parameter optimization of the pseudo mean and the inner is a standard finite-horizon MDP with an augmented state space by adding an auxiliary state of accumulated rewards. We further study the properties of this bilevel MDP, including the optimality of history-dependent deterministic policies and the piecewise quadratic concavity of the inner MDPs' optimal values with respect to the pseudo mean. To efficiently solve this bilevel MDP, we propose an iterative algorithm that alternatingly updates the inner optimal policy and the outer pseudo mean. We prove that this algorithm converges to a local optimum. We also derive a sufficient condition under which our algorithm converges to the global optimum. Furthermore, we apply this approach to study the mean-variance optimization of multi-period portfolio selection problem, which shows that our approach exactly coincides with the classical result by Li and Ng (2000) in financial engineering. Our approach builds a new avenue to solve mean-variance optimization problems and has wide applicability to any problem modeled by MDPs, which is further demonstrated by examples of mean-variance optimization for queueing control and inventory management.