TY - JOUR
T1 - FASTER LAGRANGIAN-BASED METHODS IN CONVEX OPTIMIZATION
AU - Sabach, Shoham
AU - Teboulle, Marc
N1 - Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics
PY - 2022
Y1 - 2022
N2 - In this paper, we aim at unifying, simplifying, and improving the convergence rate analysis of Lagrangian-based methods for convex optimization problems. We first introduce the notion of nice primal algorithmic map, which plays a central role in the unification and in the simplification of the analysis of most Lagrangian-based methods. Equipped with a nice primal algorithmic map, we then introduce a versatile generic scheme, which allows for the design and analysis of faster Lagrangian (FLAG) methods with new provably sublinear rate of convergence expressed in terms of function values and feasibility violation of the original (nonergodic) generated sequence. To demonstrate the power and versatility of our approach and results, we show that most well-known iconic Lagrangian-based schemes admit a nice primal algorithmic map and hence share the new faster rate of convergence results within their corresponding FLAG.
AB - In this paper, we aim at unifying, simplifying, and improving the convergence rate analysis of Lagrangian-based methods for convex optimization problems. We first introduce the notion of nice primal algorithmic map, which plays a central role in the unification and in the simplification of the analysis of most Lagrangian-based methods. Equipped with a nice primal algorithmic map, we then introduce a versatile generic scheme, which allows for the design and analysis of faster Lagrangian (FLAG) methods with new provably sublinear rate of convergence expressed in terms of function values and feasibility violation of the original (nonergodic) generated sequence. To demonstrate the power and versatility of our approach and results, we show that most well-known iconic Lagrangian-based schemes admit a nice primal algorithmic map and hence share the new faster rate of convergence results within their corresponding FLAG.
KW - Lagrangian multiplier methods
KW - alternating direction method of multiplier
KW - augmented Lagrangian
KW - convex composite minimization
KW - fast nonergodic global rate of convergence
KW - nonsmooth optimization
KW - proximal multiplier algorithms
UR - http://www.scopus.com/inward/record.url?scp=85129993694&partnerID=8YFLogxK
U2 - 10.1137/20M1375358
DO - 10.1137/20M1375358
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AN - SCOPUS:85129993694
SN - 1052-6234
VL - 32
SP - 204
EP - 227
JO - SIAM Journal on Optimization
JF - SIAM Journal on Optimization
IS - 1
ER -