6533b7cefe1ef96bd12570b1

RESEARCH PRODUCT

An Adaptive Alternating Direction Method of Multipliers

Sedi BartzRubén CampoyHung M. Phan

subject

Control and Optimizationsignal denoisingApplied Mathematicsalternating direction method of multipliersMathematics::Optimization and Controldouglas–rachford algorithmUNESCO::CIENCIAS TECNOLÓGICASManagement Science and Operations Researchcomonotonicityweakly convex functionOptimization and Control (math.OC)47H05 47N10 47J25 49M27 65K15FOS: Mathematicsfirm thresholdingMathematics - Optimization and Control

description

AbstractThe 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.

yearjournalcountryeditionlanguage
2021-03-12
http://arxiv.org/abs/2103.07159