On the linear convergence of descent methods for convex essentially smooth minimization
On the linear convergence of descent methods for convex essentially smooth minimization
复制标题
DOI：
10.1137/0330025
复制
发表时间：
1992-03
影响因子：
2.2
通讯作者：
Z. Luo;P. Tseng
中科院分区：
文献类型：
--
作者：
Z. Luo;P. Tseng
考虑最小化问题，在多面体集合上，一个仿射映射与一个严格凸本质光滑函数的组合。给出了求解该问题的下降法的线性收敛性的一般结果。利用这一结果，建立了Goldstein、Levitin和Polyak梯度投影算法的线性收敛性，以及使用正则分裂的矩阵分裂算法。结果不要求代价函数是强凸的或最优解集是有界的。分析的关键在于用新的误差界估计可行点到最优解集的距离。
Consider the problem of minimizing, over a polyhedral set, the composition of an affine mapping with a strictly convex essentially smooth function. A general result on the linear convergence of descent methods for solving this problem is presented. By applying this result, the linear convergence of both the gradient projection algorithm of Goldstein and Levitin and Polyak, and a matrix splitting algorithm using regular splitting, is established. The results do not require that the cost function be strongly convex or that the optimal solution set be bounded. The key to the analysis lies in a new error bound for estimating the distance from a feasible point to the optimal solution set.