## 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.

### MSC:

 90C25 Convex programming 65K05 Numerical mathematical programming methods 90C06 Large-scale problems in mathematical programming 49M27 Decomposition methods 90C20 Quadratic programming

Zbl 0352.65034

### Software:

Biq Mac; DIMACS; bmrm; ConstrainedLasso; QSDPNAL; PDCO; SDPNAL+; BiqMac; OEIS; QSDP
Full Text:

### References:

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