A New Decomposition Paradigm for Graph-structured Nonlinear Programs via Message Passing

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.