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arXiv · 2511.19637

Extending Douglas-Rachford Splitting for Convex Optimization

Max Nilsson·Anton Åkerman·Pontus Giselsson
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Abstract

The Douglas-Rachford splitting method is a classical and widely used algorithm for solving monotone inclusions involving the sum of two maximally monotone operators. It was recently shown to be the unique frugal, no-lifting resolvent-splitting method that is unconditionally convergent in the general two-operator setting. In this work, we show that this uniqueness does not hold in the convex optimization case: when the operators are subdifferentials of proper, closed, convex functions, a strictly larger class of frugal, no-lifting resolvent-splitting methods is unconditionally convergent. We provide a complete characterization of all such methods in the convex optimization setting and prove that this characterization is sharp: unconditional convergence holds exactly on the identified parameter regions. These results immediately yield new families of convergent ADMM-type and Chambolle-Pock-type methods obtained through their Douglas-Rachford reformulations.

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Max Nilsson, Anton Åkerman, Pontus Giselsson. 2025-11-24. Extending Douglas-Rachford Splitting for Convex Optimization. https://arxiv.org/abs/2511.19637

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