Efficient Semidefinite Programming with Approximate ADMM

Abstract Tenfold improvements in computation speed can be brought to the alternating direction method of multipliers (ADMM) for Semidefinite Programming with virtually no decrease robustness and provable convergence simply by projecting approximately cone. Instead computing projections via “exact” eigendecompositions that scale cubically matrix size cannot warm-started, we suggest using state-of-the-art factorization-free, approximate eigensolvers, thus achieving almost quadratic scaling crucial ability warm-starting. Using a recent result from Goulart et al. (Linear Algebra Appl 594:177–192, 2020. https://doi.org/10.1016/j.laa.2020.02.014 ), are able circumvent numerical instability eigendecomposition maintain tight control on projection accuracy. This turn guarantees convergence, either solution or certificate infeasibility, ADMM algorithm. To achieve this, extend results Banjac (J Optim Theory 183(2):490–519, 2019. https://doi.org/10.1007/s10957-019-01575-y ) prove reliable infeasibility detection performed even presence approximation errors. In all considered problems SDPLIB solve few thousand iterations, our approach brings significant, up 20x, speedup without noticeable increase ADMM’s iterations.