Douglas–Rachford splitting and ADMM for pathological convex optimization

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.Keywords Douglas-Rachford splitting · Strong Duality · Pathological convex programs Mathematics Subject Classification (2010) 90C46 · 49N15 · 90C25 1 Introduction Douglas-Rachford splitting (DRS) and alternating directions method of multipliers (ADMM) are classical methods originally presented in [61,35,49,47] and [44,46], respectively. DRS and ADMM are closely related. Over the last decade, these methods have enjoyed a resurgence of popularity, as the demand to solve ever larger problems grew.DRS and ADMM have strong theoretical guarantees and empirical performance, but such results are often limited to non-pathological problems; in Ernest K. Ryu UCLA

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