Time Consistency Issue in Multi-Objective Optimization

When the conditions for applying Bellman's principle of optimality hold, the pre-committed optimal policy derived by dynamic programming at initial time is time consistent, that is, the policy remains to be optimal for any state resulted in at later stages. In multi-objective optimization with a general separable structure, the pre-committed optimal policy derived by multi-objective dynamic programming is time-consistent in efficiency, that is, the policy derived at initial time remains to be efficient for any possible state at later stages, albeit not time-consistent in general. However, when a multi-objective dynamic optimization problem is not separable in the sense of multi-objective dynamic programming, the derived pre-committed policy is not time-consistent in efficiency, as witnessed in the multi-period mean-variance portfolio selection problem studied in this paper, thus leading to some irrational decision behaviours. This revealed phenomenon recognizes the importance of the time consistency issue and calls our attentions to construct more suitable decision criteria in multi-objective optimization.