This work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators. For the projective splitting framework recently proposed by Combettes and Eckstein, we show how to replace the fundamental subproblem calculation using a backward step with one based on two forward steps. The resulting algorithms have the same kind of coordination procedure and can be implemented in the same block-iterative and highly flexible manner, but may perform backward steps on some operators and forward steps on others. Prior algorithms in the projective splitting family have used only backward steps. Forward steps can be used for any Lipschitz-continuous operators provided the stepsize is bounded by the inverse of the Lipschitz constant. If the Lipschitz constant is unknown, a simple backtracking linesearch procedure may be used. For affine operators, the stepsize can be chosen adaptively without knowledge of the Lipschitz constant and without any additional forward steps. We close the paper by empirically studying the performance of several kinds of splitting algorithms on a large-scale rare feature selection problem.

中文翻译：

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具有前向步骤的投影分裂

这项工作与寻找最大单调算子和的零的经典问题有关。对于 Combettes 和 Eckstein 最近提出的投影分裂框架，我们展示了如何使用基于两个前向步骤的后向步骤替换基本子问题计算。由此产生的算法具有相同类型的协调过程，可以以相同的块迭代和高度灵活的方式实现，但可能对某些算子执行向后步骤，对其他算子执行向前步骤。投影分裂族中的先前算法仅使用后向步骤。前向步长可用于任何 Lipschitz 连续算子，前提是步长受 Lipschitz 常数的倒数限制。如果 Lipschitz 常数未知，则可以使用简单的回溯线搜索程序。对于仿射算子，可以在不知道 Lipschitz 常数的情况下自适应地选择步长，也无需任何额外的前向步长。我们通过实证研究几种分裂算法在大规模稀有特征选择问题上的性能来结束论文。