Hilbert空间上多集合分裂可行问题的KM迭代算法

KM Iterative Algorithm for Multiple-sets Split Feasibility Problem in Hilbert Space

多集合分裂可行问题就是寻找与一族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族非空闭凸集的距离最近。分裂可行问题是一类重要的最优化问题,产生于工程实践,在医学、信号处理和图像重建等领域中有着广泛的应用。文中基于n维线性空间上求解分裂可行问题的KM迭代算法,目的是要将算法在Hilbert空间中加以推广应用。通过在Hilbert空间中运用投影压缩定理,并且利用逼近函数将多集合分裂可行问题转化为最小值问题,方便了对算法的推导证明。利用上述方法可得,多集合分裂可行问题的KM迭代算法在Hilbert空间中也有较好的收敛性。因此,可以将多集合分裂可行问题的KM迭代算法在Hilbert空间中加以推广。

The multiple- sets spilt feasibility problem requires finding a point closest to a family of closed convex sets in one space,so that its image under a linear transformation will be closest to another family of closed convex sets in the image space. The multiple- sets spilt feasibility problem is an important type of optimization problem,which is generated from engineering practice and already has been widely applied in medical science,signal processing,image reconstruction. Based on KM iterative methods for solving the multiple- sets spilt feasibility problem in Rn space,try to spread this algorithm in Hilbert Space. Using projection compression theorem and approximation function transformed the multiple- sets spilt feasibility problem into a minimum value problem,making the algorithm proving more easily. By deducing and proving,the multiple- sets spilt feasibility problem has good convergence in Hilbert Space. So the result shows that the KM iterative methods are spread in Hilbert Space perfectly.