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

TL;DR

This paper introduces a novel graph-structured decomposition framework for nonlinear programs that employs message passing, enabling efficient, convergent optimization on complex hypergraph-structured problems with theoretical guarantees.

Contribution

It presents the first convergent message-passing algorithm for loopy graphs applied to nonlinear optimization, with a rigorous, implementable, and scalable approach.

Findings

Proves convergence for convex and nonconvex objectives.

Provides explicit convergence rates based on topology and partition.

Supports graph-compliant surrogates for reduced computation and communication.

Abstract

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.