An Adaptive Alternating Direction Method of Multipliers

Sedi Bartz (), Rubén Campoy () and Hung M. Phan ()
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Sedi Bartz: University of Massachusetts Lowell
Rubén Campoy: Universitat de València
Hung M. Phan: University of Massachusetts Lowell

Journal of Optimization Theory and Applications, 2022, vol. 195, issue 3, No 12, 1019-1055

Abstract: Abstract The alternating direction method of multipliers (ADMM) is a powerful splitting algorithm for linearly constrained convex optimization problems. In view of its popularity and applicability, a growing attention is drawn toward the ADMM in nonconvex settings. Recent studies of minimization problems for nonconvex functions include various combinations of assumptions on the objective function including, in particular, a Lipschitz gradient assumption. We consider the case where the objective is the sum of a strongly convex function and a weakly convex function. To this end, we present and study an adaptive version of the ADMM which incorporates generalized notions of convexity and penalty parameters adapted to the convexity constants of the functions. We prove convergence of the scheme under natural assumptions. To this end, we employ the recent adaptive Douglas–Rachford algorithm by revisiting the well-known duality relation between the classical ADMM and the Douglas–Rachford splitting algorithm, generalizing this connection to our setting. We illustrate our approach by relating and comparing to alternatives, and by numerical experiments on a signal denoising problem.

Keywords: Alternating direction method of multipliers; Douglas–Rachford algorithm; Weakly convex function; Comonotonicity; Signal denoising; Firm thresholding; 47H05; 47N10; 47J25; 49M27; 65K15 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10957-022-02098-9

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