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

Abstract: We study finite-sum nonlinear programs whose decision variables interact locally according to a graph or hypergraph. We propose MP-Jacobi (Message Passing-Jacobi), a graph-compliant decentralized framework that couples min-sum message passing with Jacobi block updates. The (hyper)graph is partitioned into tree clusters. At each iteration, agents update in parallel by solving a cluster subproblem whose objective decomposes into (i) an intra-cluster term evaluated by a single min-sum sweep on the cluster tree (cost-to-go messages) and (ii) inter-cluster couplings handled via a Jacobi correction using neighbors' latest iterates. This design uses only single-hop communication and yields a convergent message-passing method on loopy graphs. For strongly convex objectives we establish global linear convergence and explicit rates that quantify how curvature, coupling strength, and the chosen partition affect scalability and provide guidance for clustering. To mitigate the computation and communication cost of exact message updates, we develop graph-compliant surrogates that preserve convergence while reducing per-iteration complexity. We further extend MP-Jacobi to hypergraphs; in heavily overlapping regimes, a surrogate-based hyperedge-splitting scheme restores finite-time intra-cluster message updates and maintains convergence. Experiments validate the theory and show consistent improvements over decentralized gradient baselines.