分裂可行性问题的超松弛投影算法及其强收敛性

Strong Convergence of Over Relaxation Projection Algorithms for Solving Split Feasibility Problems

分裂可行性问题(Split feasibility problem, SFP)是寻找与非空闭凸集距离最近的点,并使得该点在线性变换下的像与另一非空闭凸集的距离最近。作为一类产生于工程实践的重要优化问题,在医学、信号处理和图像重建领域中被广泛应用。本文在Hilbert空间中,提出一种求解分裂可行性问题的超松弛投影算法。首先在CQ算法上引入改造的Halpern迭代序列和多个参数;然后在一定条件下,证明算法的强收敛性;数值实验结果验证了提出算法的有效性。

The splitting feasible problem is to find the point closest to a non-empty closed convex set and make the image of the point under linear transformation closest to another non-empty closed convex set. Split feasible problem is an important optimization problem, arising from engineering practice, and is widely used in the fields of medicine, signal processing and image reconstruction. In this paper, an over-relaxed projection algorithm is presented for solving the split feasible problem in Hilbert space. Firstly, the modified Halpern iteration sequence and several parameters are introduced into the CQ algorithm. Then, under certain conditions, the strong convergence of the algorithm is proved. Finally, the numerical results show that the proposed algorithm is effective.