  In contrast with many other convex optimization classes, state-of-the-art
semidefinite programming solvers are yet unable to efficiently solve large
scale instances. This work aims to reduce this scalability gap by proposing a
novel proximal algorithm for solving general semidefinite programming problems.
The proposed methodology, based on the primal-dual hybrid gradient method,
allows the presence of linear inequalities without the need of adding extra
slack variables and avoids solving a linear system at each iteration. More
importantly, it does simultaneously compute the dual variables associated with
the linear constraints. The main contribution of this work is to achieve a
substantial speedup by effectively adjusting the proposed algorithm in order to
exploit the low-rank property inherent to several semidefinite programming
problems. This proposed modification is the key element that allows the
operator splitting method to efficiently scale to larger instances. Convergence
guarantees are presented along with an intuitive interpretation of the
algorithm. Additionally, an open source semidefinite programming solver, called
ProxSDP, is made available and implementation details are discussed. Case
studies are presented in order to evaluate the performance of the proposed
methodology.
