Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting (Q2191769)

The author explores two natural extension questions concerning the celebrated Douglas-Rachford splitting for proximal point algorithms and shows that it is the unique resolvent-splitting procedure for two maximally monotone operators, while a generalization of it to three such operators is impossible without avoiding the undesired effect of enlarging the problem size. Additionally, a novel resolvent-splitting method for three maximally monotone operators with provably minimal lifting that directly generalizes the Douglas-Rachford procedure is proposed. Numerical examples in signal denoising with outliers, portfolio optimization, and Poisson denoising with \(1D\) total variation illustrate the theoretical achievements.