In this paper, we present a link between preference-based and multiobjective sequential decision-making. While transforming a multiobjective problem to a preference-based one is quite natural, the other direction is a bit less obvious. We present how this transformation (from preference-based to multiobjective) can be done under the classic condition that preferences over histories can be represented by additively decomposable utilities and that the decision criterion to evaluate policies in a state is based on expectation. This link yields a new source of multiobjective sequential decision-making problems (i.e., when reward values are unknown) and justifies the use of solving methods developed in one setting in the other one.
Multiobjective MDP (MOMDP) @cite_11 is an MDP @math where the reward function is redefined as @math with @math being the number of objectives. The value function @math of a policy @math is now vectorial and can be computed as the limit of the vectorial version of ) and ): @math , In MOMDP, the value function of policy @math Pareto-dominates that of another policy @math if in every state @math , @math is not smaller than @math on every objective and @math is greater than @math on at least one objective. By extension, we say that @math Pareto-dominates @math if value function @math Pareto-dominates value function @math . A value function (resp. policy) is Pareto-optimal if it is not Pareto-dominated by any other value function (resp. policy). Due to incomparability of vectorial value functions, there are generally many Pareto-optimal value functions (and therefore policies), which constitutes the main difficulty of the multiobjective setting.
@cite_11
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Decomposition-based algorithms have become increasingly popular for evolutionary multiobjective optimization. However, the effect of scalarizing methods used in these algorithms is still far from being well understood. This paper analyzes a family of frequently used scalarizing methods, the @math methods, and shows that the @math value is crucial to balance the selective pressure toward the Pareto optimal and the algorithm robustness to Pareto optimal front (PF) geometries. It demonstrates that an @math method that can maximize the search ability of a decomposition-based algorithm exists and guarantees that, given some weight, any solution along the PF can be found. Moreover, a simple yet effective method called Pareto adaptive scalarizing (PaS) approximation is proposed to approximate the optimal @math value. In order to demonstrate the effectiveness of PaS, we incorporate PaS into a state-of-the-art decomposition-based algorithm, i.e., multiobjective evolutionary algorithm based on decomposition (MOEA D), and compare the resultant MOEA D-PaS with some other MOEA D variants on a set of problems with different PF geometries and up to seven conflicting objectives. Experimental results demonstrate that the PaS is effective.