  This paper describes the Conic Operator Splitting Method (COSMO) solver, an
operator splitting algorithm for convex optimisation problems with quadratic
objective function and conic constraints. At each step the algorithm alternates
between solving a quasi-definite linear system with a constant coefficient
matrix and a projection onto convex sets. The low per-iteration computational
cost makes the method particularly efficient for large problems, e.g.
semidefinite programs that arise in portfolio optimisation, graph theory, and
robust control. Moreover, the solver uses chordal decomposition techniques and
a new clique merging algorithm to effectively exploit sparsity in large,
structured semidefinite programs. A number of benchmarks against other
state-of-the-art solvers for a variety of problems show the effectiveness of
our approach. Our Julia implementation is open-source, designed to be extended
and customised by the user, and is integrated into the Julia optimisation
ecosystem.
