ABSTRACT

The multiple-sets split feasibility problem (MSSFP) requires finding a point closet to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. Motivated by the ball-relaxed projection algorithm proposed by Yu et al. for the split feasibility problem (SFP), in this paper, we introduce ball-relaxed projection algorithms for solving the MSSFP. Instead of the level sets or half-spaces, our algorithms require computing the orthogonal projections onto closed balls. We establish weak and strong convergence of the proposed algorithms to a solution of the MSSFP. Finally, we provide preliminary numerical experiments to show the efficiency and the implementation of our method.

摘要

多集分裂可行性问题 (MSSFP) 需要在一个空间中找到一个最接近于一组闭凸集的点，使得其在线性变换下的图像将最接近图像空间中的另一组闭凸集。受 Yu 等人提出的球松弛投影算法的启发。对于分裂可行性问题（SFP），在本文中，我们介绍了用于解决 MSSFP 的球松弛投影算法。我们的算法需要计算封闭球上的正交投影，而不是水平集或半空间。我们建立了所提出算法对 MSSFP 解决方案的弱收敛和强收敛。最后，我们提供了初步的数值实验来展示我们方法的效率和实施。