A Simplified Form of Block-Iterative Operator Splitting and an Asynchronous Algorithm Resembling the Multi-Block Alternating Direction Method of Multipliers

• Published in: Journal of optimization theory and applications Vol. 173; no. 1; pp. 155 - 182
• Main Author: Eckstein, Jonathan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2017; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Computation; Convex analysis; Engineering; High performance computing; Mathematical models; Mathematics; Mathematics and Statistics; Multipliers; Operations Research/Decision Theory; Operators; Optimization; Optimization algorithms; Splitting; Studies; Theory of Computation; Convex optimization; 90C25; Asynchronous algorithm; 47H05; 47N10; 65Y05; Alternating direction method of multipliers (ADMM)
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-017-1074-7

This paper develops what is essentially a simplified version of the block-iterative operator splitting method already proposed by the author and P. Combettes, but with more general initialization conditions. It then describes one way of implementing this algorithm asynchronously under a computational model inspired by modern high-performance computing environments, which consist of interconnected nodes each having multiple processor cores sharing a common local memory. The asynchronous implementation framework is then applied to derive an asynchronous algorithm which resembles the alternating direction method of multipliers with an arbitrary number of blocks of variables. Unlike earlier proposals for asynchronous variants of the alternating direction method of multipliers, the algorithm relies neither on probabilistic control nor on restrictive assumptions about the problem instance, instead making only standard convex-analytic regularity assumptions. It also allows the proximal parameters to range freely between arbitrary positive bounds, possibly varying with both iterations and subproblems.