In a recent paper, Bauschke et al. study \(\rho \)-comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent \(J_A.\) In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for \(\rho \)-comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. \(zer\, A\) we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for \(\rho \)-comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability.

在最近的一篇论文中，鲍什克等人。研究\（\ RHO \） -comonotonicity作为集值运营商的单调性的广义概念，一个在Hilbert空间和描述这个条件一在其解决方法的averagedness而言\（J_A。\）在这份说明中，我们表明，该结果使得有可能使近点算法PPA及其强收敛的Halpern型变体HPPA的许多性能证明适用于此类更一般的算子。这也适用于关于收敛或亚稳率的定量结果（在T. Tao的意义上）。例如，使用这种方法，我们得到了\（\ rho \）的有限紧凑情况下PPA收敛的简单证明。-comonotone运算符，并获得有效的亚稳率。如果A具有规则模数wrt \（zer \，A \），即使没有任何紧凑性假设，我们也会得到收敛到A的零值的速率。我们还研究了\（\ rho \）- comonotone算子的PPA的Halpern型变体HPPA ，证明了其强收敛性（没有任何紧凑性或正则性假设）并给出了亚稳率。