Single-Forward-Step Projective Splitting: Exploiting Cocoercivity

Abstract

This work describes a new variant of projective splitting for monotone inclusions, in which cocoercive operators can be processed with a single forward step per iteration. This result establishes a symmetry between projective splitting algorithms, the classical forward-backward splitting method (FB), and Tseng's forward-backward-forward method (FBF). Another symmetry is that the new procedure allows for larger stepsizes for cocoercive operators: the stepsize bound is 2β for a β-cocoercive operator, which is the same as for FB. To complete the connection, we show that FB corresponds to an unattainable boundary case of the parameters in the new procedure. Unlike FB, the new method allows for a backtracking procedure when the cocoercivity constant is unknown. Proving convergence of the algorithm requires some departures from the usual proof framework for projective splitting. We close with some computational tests establishing competitive performance for the method.

Shadow Douglas–Rachford Splitting for Monotone Inclusions

“…Despite substantial progress in monotone operator theory, there are not so many original splitting algorithms for solving monotone inclusions of form (5) which use forward evaluations of B. Tseng's forward-backward-forward algorithm [24], published in 2000, was the first such method capable of solving (5). Until recently, this was the only known method with these properties, however there has been progress in the area with the discovery of further methods having this property [16,17,20]. In this connection, see also [8,12].…”