Efficient Semidefinite Programming with approximate ADMM

Tenfold speedups can be brought to ADMM for Semidefinite Programming with virtually no decrease in robustness and provable convergence simply by projecting approximately to the Semidefinite cone. Instead of computing the projections via "exact" eigendecompositions that scale cubically with the matrix size and cannot be warm-started, we suggest using state-of-the-art factorization-free, approximate eigensolvers thus achieving almost quadratic scaling and the crucial ability of warm-starting. Using a recent result from [Goulart et al., 2019] we are able to circumvent the numerically instability of the eigendecomposition and thus maintain a tight control on the projection accuracy, which in turn guarranties convergence, either to a solution or a certificate of infeasibility, of the ADMM algorithm. To achieve this, we extend recent results from [Banjac et al., 2017] to prove that reliable infeasibility detection can be performed with ADMM even in the presence of approximation errors. In all of the considered problems of SDPLIB that "exact" ADMM can solve in a few thousand iterations, our approach brings a significant, up to 20x, speedup without a noticable increase on ADMM's iterations. Further numerical results underline the robustness and efficiency of the approach