Time-Varying Trade-Offs In Multiobjective Dynamic Programming

We consider in this paper multiobjective dynamic optimization problems with a general separable structure in the sense of multiobjective dynamic programming. Different performance indices may take different functional forms and the composite operations in an individual performance index may vary from stage to stage. When some performance indices are not of a stagewise additive form, an attempt of scalarization to convert the original multiobjective dynamic optimization problem into a scalar-valued dynamic optimization problem will lead to a formulation that is nonseparable in the sense of dynamic programming. The vector-valued cost-to-go has to be recorded in order to solve the multiobjective dynamic optimization problem recursively. One important finding is that unless the vector-valued objective function is of a stagewise additive form, the optimal trade-offs among various performance indices are time-varying as the system evolves along a noninferior trajectory. The time-varying trade-offs in multiobjective dynamic programming are history-dependent. Its impact on multiobjective sequential decision-making need to be further investigated. Methods in convexifying a noninferior frontier are discussed to facilitate the solution process and to acquire the trade-off information in a most convenient way.