Decision scenarios far from rare in practice often place data-driven decision-making in an awkward position, where the decision maker may have access to neither the target distribution itself nor any empirical samples drawn from it, especially when decisions must be made in novel or highly uncertain environments. In response, robust out-of-distribution stochastic optimization (RooDSO) has emerged as a new data-driven decision-making framework of significant practical relevance, which views the unknown target distribution as an independent realization of a meta-distribution and makes robust decisions using observations from the distribution clusters related to the unknown distribution. Notably, human knowledge about the unknown typically involves both positive and negative aspects: beyond identifying the inclusive distribution clusters with which the target distribution is compatible, one can often also recognize the exclusive clusters with which it is incompatible, and the latter are sometimes even easier to obtain than the former, and thus should likewise be exploited to inform decision-making. This paper formalizes this novel RooDSO decision scenario by incorporating both inclusive and exclusive distribution clusters and establishes a complete mathematical model for it. Since these clusters are typically multi-source and heterogeneous, we integrate multiple kernel learning into the decision framework to deeply extract useful information from both sides, and establish statistical guarantees for the proposed model together with efficient learning and optimization strategies. Moreover, the proposed model allows decision makers, by tuning hyperparameters, to flexibly incorporate into the decision process their emphasis on positive versus negative information, the degree of data heterogeneity, and their risk preferences.