  In this paper, we propose a low-rank coordinate descent approach to
structured semidefinite programming with diagonal constraints. The approach,
which we call the Mixing method, is extremely simple to implement, has no free
parameters, and typically attains an order of magnitude or better improvement
in optimization performance over the current state of the art. We show that the
method is strictly decreasing, converges to a critical point, and further that
for sufficient rank all non-optimal critical points are unstable. Moreover, we
prove that with a step size, the Mixing method converges to the global optimum
of the semidefinite program almost surely in a locally linear rate under random
initialization. This is the first low-rank semidefinite programming method that
has been shown to achieve a global optimum on the spherical manifold without
assumption. We apply our algorithm to two related domains: solving the maximum
cut semidefinite relaxation, and solving a maximum satisfiability relaxation
(we also briefly consider additional applications such as learning word
embeddings). In all settings, we demonstrate substantial improvement over the
existing state of the art along various dimensions, and in total, this work
expands the scope and scale of problems that can be solved using semidefinite
programming methods.
