High-level loop transformations are a key instrument in mapping computational kernels to effectively exploit resources in modern processor architectures. However, determining appropriate compositions of loop transformations to achieve this remains a significantly challenging task; current compilers may achieve significantly lower performance than hand-optimized programs. To address this fundamental challenge, we first present a convex characterization of all distinct, semantics-preserving, multidimensional affine transformations. We then bring together algebraic, algorithmic, and performance analysis results to design a tractable optimization algorithm over this highly expressive space. The framework has been implemented and validated experimentally on a representative set of benchmarks run on state-of-the-art multi-core platforms.