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On the equivalence of inexact proximal ALM and ADMM for a class of convex composite programming. (English) Zbl 1458.90509
Summary: In this paper, we show that for a class of linearly constrained convex composite optimization problems, an (inexact) symmetric Gauss-Seidel based majorized multi-block proximal alternating direction method of multipliers (ADMM) is equivalent to an inexact proximal augmented Lagrangian method. This equivalence not only provides new perspectives for understanding some ADMM-type algorithms but also supplies meaningful guidelines on implementing them to achieve better computational efficiency. Even for the two-block case, a by-product of this equivalence is the convergence of the whole sequence generated by the classic ADMM with a step-length that exceeds the conventional upper bound of \((1+\sqrt{5})/2\), if one part of the objective is linear. This is exactly the problem setting in which the very first convergence analysis of ADMM was conducted by D. Gabay and B. Mercier [Comput. Math. Appl. 2, 17–40 (1976; Zbl 0352.65034)], but, even under notably stronger assumptions, only the convergence of the primal sequence was known. A collection of illustrative examples are provided to demonstrate the breadth of applications for which our results can be used. Numerical experiments on solving a large number of linear and convex quadratic semidefinite programming problems are conducted to illustrate how the theoretical results established here can lead to improvements on the corresponding practical implementations.
90C25 Convex programming
65K05 Numerical mathematical programming methods
90C06 Large-scale problems in mathematical programming
49M27 Decomposition methods
90C20 Quadratic programming
Full Text: DOI
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