  This paper is devoted to the design of an efficient and convergent
{semi-proximal} alternating direction method of multipliers (ADMM) for finding
a solution of low to medium accuracy to convex quadratic conic programming and
related problems. For this class of problems, the convergent two block
semi-proximal ADMM can be employed to solve their primal form in a
straightforward way. However, it is known that it is more efficient to apply
the directly extended multi-block semi-proximal ADMM, though its convergence is
not guaranteed, to the dual form of these problems. Naturally, one may ask the
following question: can one construct a convergent multi-block semi-proximal
ADMM that is more efficient than the directly extended semi-proximal ADMM?
Indeed, for linear conic programming with 4-block constraints this has been
shown to be achievable in a recent paper by Sun, Toh and Yang [arXiv preprint
arXiv:1404.5378, (2014)]. Inspired by the aforementioned work and with the
convex quadratic conic programming in mind, we propose a Schur complement based
convergent semi-proximal ADMM for solving convex programming problems, with a
coupling linear equality constraint, whose objective function is the sum of two
proper closed convex functions plus an arbitrary number of convex quadratic or
linear functions. Our convergent semi-proximal ADMM is particularly suitable
for solving convex quadratic semidefinite programming (QSDP) with constraints
consisting of linear equalities, a positive semidefinite cone and a simple
convex polyhedral set. The efficiency of our proposed algorithm is demonstrated
by numerical experiments on various examples including QSDP.
