This paper proposes an algorithm for solving structured optimization problems, which covers both the backward–backward and the Douglas–Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the corresponding operator is characterized in several cases. Convergence criteria of the algorithm in terms of general fixed point iterations are established. When applied to nonconvex feasibility including potentially inconsistent problems, we prove local linear convergence results under mild assumptions on regularity of individual sets and of the collection of sets. In this special case, we refine known linear convergence criteria for the Douglas–Rachford (DR) algorithm. As a consequence, for feasibility problem with one of the sets being affine, we establish criteria for linear and sublinear convergence of convex combinations of the alternating projection and the DR methods. These results seem to be new. We also demonstrate the seemingly improved numerical performance of this algorithm compared to the RAAR algorithm for both consistent and inconsistent sparse feasibility problems.

中文翻译：

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Douglas-Rachford算法的收敛松弛。

本文提出了一种解决结构优化问题的算法，该算法涵盖了特例的后向和后向算法以及道格拉斯-拉奇福德算法，并分析了其收敛性。相应算子的不动点集在几种情况下都有特征。建立了一般定点迭代算法的收敛准则。当应用于包括潜在不一致问题在内的非凸可行性时，我们在对单个集合和集合的集合的规律性的温和假设下证明了局部线性收敛结果。在这种特殊情况下，我们改进了道格拉斯-拉奇福德（DR）算法的已知线性收敛准则。结果，对于其中一组是仿射的可行性问题，我们建立了交替投影和DR方法的凸组合的线性和亚线性收敛准则。这些结果似乎是新的。我们还证明了与RAAR算法相比，该算法在一致和不一致的稀疏可行性问题上似乎都有改善的数值性能。