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While semidefinite programming (SDP) has traditionally been limited tomoderate-sized problems, recent algorithms augmented with matrix sketchingtechniques have enabled solving larger SDPs. However, these methods achievescalability at the cost of an increase in the number of necessary iterations,resulting in slower convergence as the problem size grows. Furthermore, theyrequire iteration-dependent parameter schedules that prohibit effectiveutilization of warm-start initializations important in practical applicationswith incrementally-arriving data or mixed-integer programming. We presentSpecBM, a provably correct, fast and scalable algorithm for solving massiveSDPs that can leverage a warm-start initialization to further accelerateconvergence. Our proposed algorithm is a spectral bundle method for solvinggeneral SDPs containing both equality and inequality constraints. Moveover,when augmented with an optional matrix sketching technique, our algorithmachieves the dramatically improved scalability of previous work whilesustaining convergence speed. We empirically demonstrate the effectiveness ofour method, both with and without warm-starting, across multiple applicationswith large instances. For example, on a problem with 600 million decisionvariables, SpecBM achieved a solution of standard accuracy in less than 7minutes, where the previous state-of-the-art scalable SDP solver requires morethan 16 hours. Our method solves an SDP with more than 10^13 decision variableson a single machine with 16 cores and no more than 128GB RAM; the previousstate-of-the-art method had not achieved an accurate solution after 72 hours onthe same instance. We make our implementation in pure JAX publicly available.

Fast, Scalable, Warm-Start Semidefinite Programming with Spectral Bundling and Sketching