  In this paper, we propose an inexact multi-block ADMM-type first-order method
for solving a class of high-dimensional convex composite conic optimization
problems to moderate accuracy. The design of this method combines an inexact
2-block majorized semi-proximal ADMM and the recent advances in the inexact
symmetric Gauss-Seidel (sGS) technique for solving a multi-block convex
composite quadratic programming whose objective contains a nonsmooth term
involving only the first block-variable. One distinctive feature of our
proposed method (the sGS-imsPADMM) is that it only needs one cycle of an
inexact sGS method, instead of an unknown number of cycles, to solve each of
the subproblems involved.With some simple and implementable error tolerance
criteria, the cost for solving the subproblems can be greatly reduced, and many
steps in the forward sweep of each sGS cycle can often be skipped, which
further contributes to the efficiency of the proposed method. Global
convergence as well as the iteration complexity in the non-ergodic sense is
established.Preliminary numerical experiments on some high-dimensional linear
and convex quadratic SDP problems with a large number of linear equality and
inequality constraints are also provided. The results show that for the vast
majority of the tested problems, the sGS-imsPADMM is 2 to 3 times faster than
the directly extended multi-block ADMM with the aggressive step-length of
1.618, which is currently the benchmark among first-order methods for solving
multi-block linear and quadratic SDP problems though its convergence is not
guaranteed.
