Index tuning, i.e., selecting the indexes appropriate for a workload, is a crucial problem in database system tuning. In this paper, we solve index tuning for large problem instances that are common in practice, e.g., thousands of queries in the workload, thousands of candidate indexes and several hard and soft constraints. Our work is the first to reveal that the index tuning problem has a well structured space of solutions, and this space can be explored efficiently with well known techniques from linear optimization. Experimental results demonstrate that our approach outperforms state-of-the-art commercial and research techniques by a significant margin (up to an order of magnitude).