Name: Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting
Creator: Ernest K. Ryu
Keywords: first-order methods, Numerical optimization and variational techniques, Convex programming, splitting methods, 0211 other engineering and technologies, 02 engineering and technology, 16. Peace & justice, 01 natural sciences, Numerical methods involving duality, lower bounds

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 22 May 2019Embargo end date: 01 Jan 2018 English Publisher:Springer Science and Business Media LLCJournal:Mathematical Programming, volume 182, pages 233-273 (issn: 0025-5610, eissn: 1436-4646,

Authors: Ernest K. Ryu;

doi: 10.1007/s10107-019-01403-1 , 10.48550/arxiv.1802.07534

arXiv: 1802.07534

Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting

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Abstract

Given the success of Douglas--Rachford splitting (DRS), it is natural to ask whether DRS can be generalized. Are there other 2 operator resolvent-splittings sharing the favorable properties of DRS? Can DRS be generalized to 3 operators? This work presents the answers: no and no. In a certain sense, DRS is the unique 2 operator resolvent-splitting, and generalizing DRS to 3 operators is impossible without lifting, where lifting roughly corresponds to enlarging the problem size. The impossibility result further raises a question. How much lifting is necessary to generalize DRS to 3 operators? This work presents the answer by providing a novel 3 operator resolvent-splitting with provably minimal lifting that directly generalizes DRS.

Published in Mathematical Programming (updated version with corrected typo)

Related Organizations

University of California, Los Angeles
United States

Keywords

first-order methods, Numerical optimization and variational techniques, Convex programming, splitting methods, Numerical methods involving duality, lower bounds, Iterative procedures involving nonlinear operators, Optimization and Control (math.OC), Douglas-Rachford splitting, Optimization and Control, FOS: Mathematics, Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc., maximal monotone operators, Monotone operators and generalizations

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