Second-Order KKT Guarantees for Bregman ADMM in Nonconvex Optimization (arXiv 2606.28307)

Summary

A paper by Shuang Li, Zhihui Zhu, and Qiuwei Li (arXiv 2606.28307, posted June 2026) analyzes Bregman ADMM for nonconvex linearly constrained problems under two-sided relative smoothness, which replaces the standard Lipschitz gradient assumption with a Hessian comparison relative to a Bregman kernel. This setting covers polynomial objectives arising in matrix and tensor models where a global Lipschitz-gradient constant may not exist. The authors show that on an invariant open state-space domain, one iteration of Bregman ADMM defines a smooth primal-dual fixed-point map whose strict-saddle KKT points are unstable fixed points. Consequently, with random initialization, the iterates converge to a strict saddle with probability zero. Combined with existing first-order convergence results, this yields almost-sure second-order stationarity of limiting KKT points. The analysis extends to multi-block star-consensus distributed optimization, with numerical experiments on distributed matrix factorization and an example on symmetric tensor factorization demonstrating the broader Bregman proximal splitting framework.