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

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 with O(1/k) ergodic rate for its primal-dual gap function. 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.