  While semidefinite programming (SDP) has traditionally been limited to
moderate-sized problems, recent algorithms augmented with matrix sketching
techniques have enabled solving larger SDPs. However, these methods achieve
scalability at the cost of an increase in the number of necessary iterations,
resulting in slower convergence as the problem size grows. Furthermore, they
require iteration-dependent parameter schedules that prohibit effective
utilization of warm-start initializations important in practical applications
with incrementally-arriving data or mixed-integer programming. We present
Unified Spectral Bundling with Sketching (USBS), a provably correct, fast and
scalable algorithm for solving massive SDPs that can leverage a warm-start
initialization to further accelerate convergence. Our proposed algorithm is a
spectral bundle method for solving general SDPs containing both equality and
inequality constraints. Moveover, when augmented with an optional matrix
sketching technique, our algorithm achieves the dramatically improved
scalability of previous work while sustaining convergence speed. We empirically
demonstrate the effectiveness of our method across multiple applications, with
and without warm-starting. For example, USBS provides a 500x speed-up over the
state-of-the-art scalable SDP solver on an instance with over 2 billion
decision variables.
