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The objective of this paper is to design an efficient and convergentalternating direction method of multipliers (ADMM) for finding a solution ofmedium accuracy to conic programming problems whose constraints consist oflinear equalities, linear inequalities, a non-polyhedral cone and a polyhedralcone. For this class of problems, one may apply the directly extended ADMM totheir dual, which can be written in the form of convex programming with fourseparable blocks in the objective function and a coupling linear equationconstraint. Indeed, the directly extended ADMM, though may diverge in theory,often performs much better numerically than many of its variants withtheoretical convergence guarantee. Ideally, one should find a convergentvariant which is at least as efficient as the directly extended ADMM inpractice. We achieve this goal by designing a convergent semi-proximal ADMM(called sPADMM3c for convenience) for convex programming problems having threeseparable blocks in the objective function with the third part being linear. Ateach iteration, the proposed sPADMM3c takes one special block coordinatedescent (BCD) cycle with the order $1 \rightarrow 3 \rightarrow 2 \rightarrow3$, instead of the usual $1 \rightarrow 2 \rightarrow 3$ Gauss-Seidel BCD cycleused in the non-convergent directly extended $3$-block ADMM, for updating thevariable blocks. Our extensive numerical tests on the important class of doublynon-negative semidefinite programming (SDP) problems with linear equalityand/or inequality constraints demonstrate that our convergent method is atleast $20%$ faster than the directly extended ADMM with unit step-length forthe vast majority of about $550$ large scale problems tested.

A Convergent 3-Block Semi-Proximal Alternating Direction Method of Multipliers for Conic Programming with $4$-Type of Constraints