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This work describes a new variant of projective splitting for solving maximalmonotone inclusions and complicated convex optimization problems. In the newversion, cocoercive operators can be processed with a single forward step periteration. In the convex optimization context, cocoercivity is equivalent toLipschitz differentiability. Prior forward-step versions of projectivesplitting did not fully exploit cocoercivity and required two forward steps periteration for such operators. Our new single-forward-step method establishes asymmetry between projective splitting algorithms, the classicalforward-backward splitting method (FB), and Tseng's forward-backward-forwardmethod (FBF). The new procedure allows for larger stepsizes for cocoerciveoperators: 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 toan unattainable boundary case of the parameters in the new procedure. UnlikeFB, the new method allows for a backtracking procedure when the cocoercivityconstant is unknown. Proving convergence of the algorithm requires somedepartures from the prior proof framework for projective splitting. We closewith some computational tests establishing competitive performance for themethod.

Single-Forward-Step Projective Splitting: Exploiting Cocoercivity