  Motivated by the task of clustering either $d$ variables or $d$ points into
$K$ groups, we investigate efficient algorithms to solve the Peng-Wei (P-W)
$K$-means semi-definite programming (SDP) relaxation. The P-W SDP has been
shown in the literature to have good statistical properties in a variety of
settings, but remains intractable to solve in practice. To this end we propose
FORCE, a new algorithm to solve this SDP relaxation. Compared to the naive
interior point method, our method reduces the computational complexity of
solving the SDP from $\tilde{O}(d^7\log\epsilon^{-1})$ to
$\tilde{O}(d^{6}K^{-2}\epsilon^{-1})$ arithmetic operations for an
$\epsilon$-optimal solution. Our method combines a primal first-order method
with a dual optimality certificate search, which when successful, allows for
early termination of the primal method. We show for certain variable clustering
problems that, with high probability, FORCE is guaranteed to find the optimal
solution to the SDP relaxation and provide a certificate of exact optimality.
As verified by our numerical experiments, this allows FORCE to solve the P-W
SDP with dimensions in the hundreds in only tens of seconds. For a variation of
the P-W SDP where $K$ is not known a priori a slight modification of FORCE
reduces the computational complexity of solving this problem as well: from
$\tilde{O}(d^7\log\epsilon^{-1})$ using a standard SDP solver to
$\tilde{O}(d^{4}\epsilon^{-1})$.
