Uniqueness of DRS as the 2 Operator Resolvent-Splitting and Impossibility of 3 Operator Resolvent-Splitting

TL;DR

This work presents the answer by providing a novel 3 operator resolvent-splitting with provably minimal lifting that directly generalizes DRS.

Abstract

Given the success of Douglas--Rachford splitting (DRS), it is natural to ask whether DRS can be generalized. Are there other 2 operator resolvent-splittings sharing the favorable properties of DRS? Can DRS be generalized to 3 operators? This work presents the answers: no and no. In a certain sense, DRS is the unique 2 operator resolvent-splitting, and generalizing DRS to 3 operators is impossible without lifting, where lifting roughly corresponds to enlarging the problem size. The impossibility result further raises a question. How much lifting is necessary to generalize DRS to 3 operators? This work presents the answer by providing a novel 3 operator resolvent-splitting with provably minimal lifting that directly generalizes DRS.

Figures (3)

• Figure 1: Objective value and $|f(x^{k+1})-f(x^{k})|/f(x^{k})$ vs. iterations for the denoising problem.
• Figure 2: $|$Objective value suboptimality$|$, $|f(x^{k+1})-f(x^{k})|/f(x^{k})$, and distance to solution vs. iterations for the portfolio optimization problem. We take the absolute value in the first plot, because the slightly infeasible iterates produce objective values lower than the optimal value. The rough cost per iteration is $0.025s$ for CV and DYS and $0.15s$ for the splitting of Theorem \ref{['thm:attainment']}, PPXA, and Bot--Hendrich.
• Figure 3: Objective value, $|(f(x^{k+1})-f(x^{k}))/f(x^{k})|$, and distance to solution vs. iterations for the Poisson denoising with 1D total variation problem.

Theorems & Definitions (7)

• theorem 1
• theorem 2
• corollary thmcountercorollary
• proof
• lemma thmcounterlemma
• theorem 3
• theorem 4