Thao Nguyen Hieu
1Delft Center for Systems and Control, Delft University of Technology, 2628CD Delft, The Netherlands.
2Department of Mathematics, School of Education, Can Tho University, Can Tho, Vietnam.
Comput Optim Appl. 2018;70(3):841-863. doi: 10.1007/s10589-018-9989-y. Epub 2018 Mar 6.
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.
本文提出了一种用于求解结构化优化问题的算法，该算法涵盖了向后 - 向后算法和道格拉斯 - 拉赫福德算法这两种特殊情况，并分析了其收敛性。在几种情况下对相应算子的不动点集进行了刻画。建立了基于一般不动点迭代的算法收敛准则。当应用于包括潜在不一致问题的非凸可行性问题时，我们在关于各个集合和集合族正则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，我们细化了道格拉斯 - 拉赫福德（DR）算法的已知线性收敛准则。因此，对于其中一个集合为仿射集的可行性问题，我们建立了交替投影法和DR方法的凸组合的线性和次线性收敛准则。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稀疏可行性问题上的数值性能似乎有所提高。
