A Family of Projection Gradient Methods for Solving the Multiple-Sets Split Feasibility Problem
A Family of Projection Gradient Methods for Solving the Multiple-Sets Split Feasibility Problem
复制标题
求解多集分割可行性问题的一族投影梯度法
DOI：
10.1007/s10957-019-01563-2
复制
发表时间：
2019
影响因子：
1.9
通讯作者：
Xiaojun Zhuang
中科院分区：
文献类型：
--
作者：
Jinhua Wang;Yaohua Hu;Carisa Kwok Wai Yu;Xiaojun Zhuang
在本文中，我们探索一族用于求解多集合分裂可行性问题的投影梯度方法，其包括在Wen等人（J Optim Theory Appl 166：844 - 860，2015）中引入的循环/同时迭代方法作为特例。对于一般情况下，其中所涉及的集是由凸函数的水平集，在水平集上的投影的计算一般是复杂的，因此，所得到的投影梯度方法不能容易地实现。为了避免这一困难，我们引入了一个家庭的放松投影梯度方法，在近似半空间的投影采用的水平集上的地方。它们涵盖了在Wen等人（J Optim Theory Appl 166：844 - 860，2015）中引入的松弛循环/同时迭代方法作为特例。建立了这些方法的全局弱收敛定理。特别地，作为已建立定理的直接应用，我们的结果填补了一些空白，并处理了Wen等人（J Optim Theory Appl 166：844 - 860，2015）中出现的不完善之处，从而改进和扩展了其中的相应结果。
In the present paper, we explore a family of projection gradient methods for solving the multiple-sets split feasibility problem, which include the cyclic/simultaneous iteration methods introduced in Wen et al. (J Optim Theory Appl 166:844–860, 2015) as special cases. For the general case, where the involved sets are given by level sets of convex functions, the calculation of the projection onto the level sets is complicated in general, and thus, the resulting projection gradient method cannot be implemented easily. To avoid this difficulty, we introduce a family of relaxed projection gradient methods, in which the projections onto the approximated halfspaces are adopted in place of the ones onto the level sets. They cover the relaxed cyclic/simultaneous iteration methods introduced in Wen et al. (J Optim Theory Appl 166:844–860, 2015) as special cases. Global weak convergence theorems are established for these methods. In particular, as direct applications of the established theorems, our results fill some gaps and deal with the imperfections that appeared in Wen et al. (J Optim Theory Appl 166:844–860, 2015) and hence improve and extend the corresponding results therein.