A frugal primal-dual splitting with minimal lifting over arbitrary rooted trees

Abstract

We develop a frugal primal-dual splitting with minimal lifting for solving structured monotone inclusions, involving cocoercive operators, linear compositions and parallel sums. This is established by defining hierarchical nodes and edges between them over a tree-structured graph, with arbitrary assignments of dual variables and cocoercive elements to primal nodes. This arbitrariness allows a great flexibility in terms of level-synchronous distributed computing, such as centralized or decentralized. The particular instances naturally extend the Douglas--Rachford splitting on various graphs, recover the parallel Chambolle--Pock, and solve a class of structured convex minimization problems. For the pure resolvent convex minimization subclass, we establish an O(1/k) ergodic rate for a restricted primal-dual gap over bounded test sets. Furthermore, we introduce a reformulation technique in a product Hilbert space to facilitate the convergence analysis, specifically to derive the o(1/k)-rate of asymptotic regularity

Disclosure

“ted to Prof. Yuchao Tang (Guangzhou University) for sharing his preprint [29] with us, and to Prof. Xiaokai Chang (Lanzhou University of Technology) for his valuable comments. We did not use AI to develop the ideas or proofs in this paper. AI tools were used to check grammar and typos. REFERENCES [1] A. Åkerman, P. Chenchene, E. an Giselsson, and E. Naldi, Splitting the forward-backward algorithm: A full characterization, Arxiv pr”

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2026-08-23T19:01:04.318529Z

Structural counts