Hilbert 空间上多集合分裂可行问题的KM 迭代算法

摘要

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 wide-ly 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.