Second-Order KKT Guarantees for Bregman ADMM in Nonconvex and Non-Lipschitz Optimization

2026-06-26

AI Summary

This work addresses the lack of second-order convergence guarantees for Bregman alternating direction method of multipliers (Bregman ADMM) in linearly constrained optimization with nonconvex and non-Lipschitz gradients. Under a two-sided relative smoothness assumption, the authors construct a smooth primal–dual fixed-point mapping and prove that KKT points corresponding to strict saddles are unstable fixed points, thereby ensuring that iterates initialized randomly converge almost surely to second-order stationary KKT points. The study introduces two key innovations: a symmetrized scaling of the Bregman kernel and a null-space cancellation technique tailored to star-structured graphs. These enable the first second-order convergence theory for Bregman ADMM in non-Lipschitz, nonconvex settings. Numerical experiments on distributed matrix and symmetric tensor decomposition demonstrate the efficacy of the proposed approach.

Abstract

We analyze Bregman ADMM for nonconvex linearly constrained problems under two-sided relative smoothness, a condition that 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 for which a global Lipschitz-gradient constant need not exist. We 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, from 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. We extend the analysis to a multi-block star consensus formulation for distributed optimization. The technical novelty lies in a determinant reduction with a Bregman-specific symmetrization and scaling step in the two block spectral argument, together with a null space cancellation exploiting the star graph structure in the consensus case. Numerical experiments on distributed matrix factorization illustrate the theory, and a symmetric tensor factorization example demonstrates the broader Bregman proximal splitting idea beyond the separable consensus setting.

Problem

• nonconvex optimization
• non-Lipschitz
• Bregman ADMM
• second-order KKT
• strict saddle

Innovation

• Bregman ADMM
• second-order KKT
• nonconvex optimization
• relative smoothness
• distributed consensus