Efficient Semidefinite Programming with approximate ADMM

Understand

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., 2020], 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.