Single-Forward-Step Projective Splitting: Exploiting Cocoercivity
This work provides an incremental improvement for researchers in optimization by enhancing projective splitting algorithms to better exploit cocoercivity, aligning them more closely with classical methods like forward-backward splitting.
The authors tackled the problem of solving maximal monotone inclusions and complex convex optimization by introducing a new projective splitting variant that processes cocoercive operators with a single forward step per iteration, achieving a stepsize bound of 2β for β-cocoercive operators and competitive performance in computational tests.
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 (FBF). The new procedure allows for larger stepsizes for cocoercive operators: the stepsize bound is $2β$ for a $β$-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.