Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Programming

Deren Han, Defeng Sun, Liwei Zhang
2018 Mathematics of Operations Research  
In this paper, we aim to prove the linear rate convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex composite optimization problems. Under a mild calmness condition, which holds automatically for convex composite piecewise linear-quadratic programming, we establish the global Q-linear rate of convergence for a general semi-proximal ADMM with the dual step-length being taken in (0, (1 + √ 5)/2). This semi-proximal ADMM, which covers the
more » ... ssic one, has the advantage to resolve the potentially non-solvability issue of the subproblems in the classic ADMM and possesses the abilities of handling the multi-block cases efficiently. We demonstrate the usefulness of the obtained results when applied to two-and multi-block convex quadratic (semidefinite) programming.
doi:10.1287/moor.2017.0875 fatcat:hi4rvx7z4vforefsbro35lt5mu