In this paper we introduce the problem of Constrained Subspace Skyline Queries. This class of queries can be thought of as a generalization of subspace skyline queries using range constraints. Although both constrained skyline queries and subspace skyline queries have been addressed previously, the implications of constrained subspace skyline queries has not been examined so far. Constrained skyline queries are usually more expensive than regular skylines. In case of constrained subspace skyline queries additional performance degradation is caused through the projection. In order to support constrained skylines for arbitrary subspaces, we present approaches exploiting multiple low-dimensional indexes instead of relying on a single high-dimensional index. Effective pruning strategies are applied to discard points from dominated regions. An important ingredient of our approach is the workload-adaptive strategy for determining the number of indexes and the assignment of dimensions to the indexes. Extensive performance evaluation shows the superiority of our proposed technique compared to its most related competitors.