一种在任意根树上具有最小提升的节俭原始对偶分裂
A frugal primal-dual splitting with minimal lifting over arbitrary rooted trees
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中文总结 AI 辅助
研究解决结构化单调包含问题，提出具有最小提升的节俭原始对偶分裂方法，通过在树状图上定义节点和边实现，该方法灵活性高，能扩展相关分裂方法，解决一类凸最小化问题并推导收敛率。
中文摘要 AI 辅助
我们开发了一种具有最小提升的节俭原始对偶分裂方法，用于解决涉及余强制算子、线性组合和平行和的结构化单调包含问题。通过在树状图上定义层次节点及其间的边来实现，对偶变量和余强制元素可任意分配给原始节点。这种任意性在水平同步分布式计算方面具有很大灵活性，如集中式或分散式。特定实例自然扩展了各种图上的Douglas - Rachford分裂，恢复了并行Chambolle - Pock方法，并以O(1/k)遍历率解决一类结构化凸最小化问题的原始对偶间隙函数。此外，我们在乘积希尔伯特空间中引入一种重新表述技术以促进收敛分析，特别是推导渐近正则性的o(1/k)率。
英文摘要
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
