In a series of papers (Solodov and Svaiter in J Convex Anal 6(1):59–70, 1999; Set-Valued Anal 7(4):323–345, 1999; Numer Funct Anal Optim 22(7–8):1013–1035, 2001) Solodov and Svaiter introduced new inexact variants of the proximal point method with relative error tolerances. Point-wise and ergodic iteration-complexity bounds for one of these methods, namely the hybrid proximal extragradient method (1999) were established by Monteiro and Svaiter (SIAM J Optim 20(6):2755–2787, 2010). Here, we extend these results to a more general framework, by establishing point-wise and ergodic iteration-complexity bounds for the inexact proximal point method studied by Solodov and Svaiter (2001). Using this framework we derive global convergence results and iteration-complexity bounds for a family of projective splitting methods for solving monotone inclusion problems, which generalize the projective splitting methods introduced and studied by Eckstein and Svaiter (SIAM J Control Optim 48(2):787–811, 2009).

中文翻译：

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关于求解单调包含问题的混合近端梯度投影方法的复杂性

在一系列论文中（Solodov和Svaiter在J Convex Anal 6（1）：59-70，1999; Set-Valued Anal 7（4）：323-345，1999; Numer Funct Anal Optim 22（7-8）： 1013–1035，2001年）Solodov和Svaiter提出了具有相对误差容限的近点法的新的不精确变体。这些方法之一的点向遍历遍历复杂性边界，即混合近端超梯度方法（1999年）是由Monteiro和Svaiter建立的（SIAM J Optim 20（6）：2755–2787，2010）。在这里，通过为Solodov和Svaiter（2001）研究的不精确的近端点方法建立逐点遍历遍历和遍历遍历复杂性边界，我们将这些结果扩展到一个更通用的框架。