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We propose a new algorithm to solve optimization problems of the form $\minf(X)$ for a smooth function $f$ under the constraints that $X$ is positivesemidefinite and the diagonal blocks of $X$ are small identity matrices. Suchproblems often arise as the result of relaxing a rank constraint (lifting). Inparticular, many estimation tasks involving phases, rotations, orthonormalbases or permutations fit in this framework, and so do certain relaxations ofcombinatorial problems such as Max-Cut. The proposed algorithm exploits thefacts that (1) such formulations admit low-rank solutions, and (2) theirrank-restricted versions are smooth optimization problems on a Riemannianmanifold. Combining insights from both the Riemannian and the convex geometriesof the problem, we characterize when second-order critical points of the smoothproblem reveal KKT points of the semidefinite problem. We compare against stateof the art, mature software and find that, on certain interesting probleminstances, what we call the staircase method is orders of magnitude faster, ismore accurate and scales better. Code is available.

A Riemannian low-rank method for optimization over semidefinite matrices with block-diagonal constraints