We study finite-sum nonlinear programs with localized variable coupling encoded by a (hyper)graph. We introduce a graph-compliant decomposition framework that brings message passing into continuous optimization in a rigorous, implementable, and provable way. The (hyper)graph is partitioned into tree clusters (hypertree factor graphs). At each iteration, agents update in parallel by solving local subproblems whose objective splits into an {\it intra}-cluster term summarized by cost-to-go messages from one min-sum sweep on the cluster tree, and an {\it inter}-cluster coupling term handled Jacobi-style using the latest out-of-cluster variables. To reduce computation/communication, the method supports graph-compliant surrogates that replace exact messages/local solves with compact low-dimensional parametrizations; in hypergraphs, the same principle enables surrogate hyperedge splitting, to tame heavy hyperedge overlaps while retaining finite-time intra-cluster message updates and efficient computation/communication. We establish convergence for (strongly) convex and nonconvex objectives, with topology- and partition-explicit rates that quantify curvature/coupling effects and guide clustering and scalability. To our knowledge, this is the first convergent message-passing method on loopy graphs.

翻译：本文研究具有局部变量耦合的有限和式非线性规划问题，其耦合结构由（超）图编码。我们提出一种图相容分解框架，以严谨、可实施且可证明的方式将消息传递引入连续优化领域。该框架将（超）图划分为树状簇（超树因子图）。在每次迭代中，智能体通过并行求解局部子问题进行更新：其目标函数分解为两部分——由簇树单次最小和扫描生成的代价传递消息所汇总的簇内项，以及采用雅可比风格利用最新簇外变量处理的簇间耦合项。为降低计算/通信开销，本方法支持采用图相容代理函数，通过紧凑的低维参数化替代精确消息传递/局部求解；在超图场景中，相同原理可实现代理超边分割，既能控制密集超边重叠，又能保持有限时间内簇内消息更新与高效计算/通信。我们针对（强）凸及非凸目标函数建立了收敛性证明，并给出显式依赖拓扑与划分结构的收敛速率，该速率可量化曲率/耦合效应，为聚类设计与可扩展性提供理论指导。据我们所知，这是首个在带环图上具有收敛保证的消息传递方法。