Single-forward-step projective splitting: exploiting cocoercivity

This work describes a new variant of projective splitting for solving maximal monotone inclusions and complicated convex optimization problems. In the new version, cocoercive operators can be processed with a single forward step per iteration. In the convex optimization context, cocoercivity is equivalent to Lipschitz differentiability. Prior forward-step versions of projective splitting did not fully exploit cocoercivity and required two forward steps per iteration for such operators. Our new single-forward-step method establishes a symmetry between projective splitting algorithms, the classical forward-backward splitting method (FB), and Tseng's forward-backward-forward method. The new procedure allows for larger stepsizes for cocoercive operators: the stepsize bound is 2 beta for a beta-cocoercive operator, the same bound as has been established for FB. 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 prior proof framework for projective splitting. We close with some computational tests establishing competitive performance for the method.

Automated summary

This work describes a new variant of projective splitting that can solve maximal monotone inclusions and complicated convex optimization problems with a single forward step for processing cocoercive operators. The new method allows larger stepsizes for cocoercive operators by establishing a stepsize bound of 2 beta for a beta-cocoercive operator. The new method also allows for a backtracking procedure when the cocoercivity constant is unknown, unlike the previous forward-backward splitting method.