Convergence Rate of Inexact Proximal Point Algorithms for Operator with Hölder Metric Subregularity
研究了Hilbert空间中极大单调算子的不精确近端点算法的强收敛性，在Hölder度量次正则条件下建立了全局/局部强收敛性并给出了收敛速度的定量估计，改进了经典精确近端点算法的相关结果。
We study the issue of strong convergence of inexact proximal point algorithms (introduced by Rockafellar in [SIAM J. Control Optim., 14 (1976), pp. 877–898]) for maximal monotone operators on Hilbert spaces. A unified global/local strong convergence of inexact proximal point algorithms is established under the Hölder metrically subregular condition. Furthermore, quantitative estimates on the convergence rate of inexact proximal point algorithms are also provided. Applying to the special case of the classical (exact) proximal point algorithm, our results improve the corresponding ones in [G. Li and B. S. Mordukhovich, SIAM J. Optim., 22 (2012), pp. 1655–1684]. Finally, as applications, global/local strong convergence and estimates on the convergence rate of inexact proximal point algorithms for optimization problems are presented.