  Many problems in control theory can be formulated as semidefinite programs
(SDPs). For large-scale SDPs, it is important to exploit the inherent sparsity
to improve the scalability. This paper develops efficient first-order methods
to solve SDPs with chordal sparsity based on the alternating direction method
of multipliers (ADMM). We show that chordal decomposition can be applied to
either the primal or the dual standard form of a sparse SDP, resulting in
scaled versions of ADMM algorithms with the same computational cost. Each
iteration of our algorithms consists of a projection on the product of small
positive semidefinite cones, followed by a projection on an affine set, both of
which can be carried out efficiently. Our techniques are implemented in CDCS,
an open source add-on to MATLAB. Numerical experiments on large-scale sparse
problems in SDPLIB and random SDPs with block-arrow sparse patterns show
speedups compared to some common state-of-the-art software packages.
