Douglas--Rachford Splitting and ADMM for Pathological Convex Optimization

TL;DR

This paper investigates the behavior of Douglas--Rachford Splitting and ADMM algorithms under pathological conditions where standard assumptions like primal and dual solutions or strong duality may not hold, showing they still perform well with only strong duality.

Contribution

The paper demonstrates that DRS and ADMM algorithms require only strong duality to ensure asymptotic feasibility and optimality, extending their applicability beyond classical assumptions.

Findings

DRS and ADMM work under only strong duality.

Asymptotic feasibility and optimality are achieved without primal or dual solutions.

Performance degrades significantly when assumptions are violated.

Abstract

Despite the vast literature on DRS and ADMM, there has been very little work analyzing their behavior under pathologies. Most analyses assume a primal solution exists, a dual solution exists, and strong duality holds. When these assumptions are not met, i.e., under pathologies, the theory often breaks down and the empirical performance may degrade significantly. In this paper, we establish that DRS only requires strong duality to work, in the sense that asymptotically iterates are approximately feasible and approximately optimal.