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.

这项工作描述了射影分裂的新变体，用于解决最大单调包含和复杂的凸优化问题。在新版本中，可以在每次迭代中使用一个前进步骤来处理矫顽算子。在凸优化上下文中，矫顽力等于Lipschitz可微性。射影分裂的先前前进步骤版本没有完全利用矫顽力，并且对于此类算子，每次迭代都需要两个前进步骤。我们的新的单步执行方法在投影拆分算法，经典的前向后拆分方法（FB）和Tseng的前向后退方法之间建立了对称性。新过程允许矫顽算子具有更大的步长：步长的边界为\（2 \ beta \）对于\（\ beta \）- cocoercive运算符，其边界与为FB建立的约束相同。我们显示FB对应于新过程中参数无法达到的边界情况。与FB不同，新方法允许在矫顽力常数未知时执行回溯程序。证明算法的收敛性需要与射影分裂的先验证明框架有所不同。我们以一些计算测试结束，以建立该方法的竞争性能。