Efficient Semidefinite Programming with Approximate ADMM

Nikitas Rontsis, Paul Goulart and Yuji Nakatsukasa

@article{NGN:2022,
  author = {Rontsis, Nikitas and Goulart, Paul and Nakatsukasa, Yuji},
  title = {Efficient Semidefinite Programming with Approximate ADMM},
  journal = {Journal of Optimization Theory and Applications},
  year = {2022},
  volume = {192},
  number = {1},
  pages = {292-320},
  url = {https://doi.org/10.1007/s10957-021-01971-3},
  doi = {10.1007/s10957-021-01971-3}
}

Tenfold improvements in computation speed can be brought to the alternating direction method of multipliers (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. (Linear Algebra Appl 594:177–192, 2020. https:doi.org10.1016j.laa.2020.02.014), we are able to circumvent the numerical instability of the eigendecomposition and thus maintain tight control on the projection accuracy. This in turn guarantees 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. (J Optim Theory Appl 183(2):490–519, 2019. https:doi.org10.1007s10957-019-01575-y) 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 noticeable increase in ADMM's iterations.