A Three-Operator Splitting Scheme Derived from Three-Block ADMM
Anshika Anshika, Jiaxing Li, Debdas Ghosh, Xiangxiong Zhang
TL;DR
The paper introduces a novel three-operator splitting operator for monotone inclusions and composite convex minimization, derived from a modified dual form of the classical three-block ADMM. The operator $T$ is explicit and yields a Krasnosel’skiĭ–Mann iteration, with connections to existing schemes like Davis–Yin and PD3O; notably, it remains robust to larger step sizes and can be $1/2$-averaged under orthogonal-domain assumptions, guaranteeing convergence without stringent step-size constraints in that special case. The authors provide a rigorous convergence analysis, including weak convergence to fixed points and sublinear rates, and extend the framework to multi-operator settings. Numerical experiments demonstrate robustness and practical effectiveness, particularly when Lipschitz constants are unknown or difficult to estimate, highlighting the method’s potential for large-scale convex optimization and operator inclusion problems.
Abstract
This work presents a new three-operator splitting method to handle monotone inclusion and convex optimization problems. The proposed splitting serves as another natural extension of the Douglas-Rachford splitting technique to problems involving three operators. For solving a composite convex minimization of a sum of three functions, its formula resembles but is different from Davis-Yin splitting and the dual formulation of the classical three-block ADMM. Numerical tests suggest that such a splitting scheme is robust in the sense of allowing larger step sizes. When two functions have orthogonal domains, the splitting operator can be proven 1/2-averaged, which implies convergence of the iteration scheme using any positive step size.