Loop nest optimization is a combinatorial problem. Due to the growing complexity of modern architectures, it involves two increasingly difficult tasks: (1) analyzing the profitability of sequences of transformations to enhance parallelism, locality, and resource usage, which amounts to a hard problem on a non-linear objective function; (2) the construction and exploration of search space of legal transformation sequences. Practical optimizing and parallelizing compilers decouple these tasks, resorting to a predefined set of enabling transformations to eliminate all sorts of optimization-limiting semantical constraints. State-of-the-art optimization heuristics face a hard decision problem on the selection of enabling transformations only remotely related to performance.

We propose a new design where optimization heuristics first address the main performance anomalies, then correct potentially illegal loop transformations a posteriori, attempting to minimize the performance impact of the necessary adjustments. We propose a general method to correct any sequence of loop transformations through a combination of loop shifting, code motion and index-set splitting. Sequences of transformations are modeled by compositions of geometric transformations on multidimensional affine schedules. We provide experimental evidence of the scalability of the algorithms on real loop optimizations.