On the proximal point algorithms for solving the monotone inclusion problem

We consider finding a zero point of the maximally monotone operator $T$. First, instead of using the proximal point algorithm (PPA) for this purpose, we employ PPA to solve its Yosida regularization $T_{\lambda}$. Then, based on an $O(a_{k+1})$ ($a_{k+1}\geq \varepsilon>0$) resolvent index of $T$, it turns out that we can establish a convergence rate of $O (1/{\sqrt{\sum_{i=0}^{k}a_{i+1}^2}})$ for both the $\|T_{\lambda}(\cdot)\|$ and the gap function $\mathtt{Gap}(\cdot)$ in the non-ergodic sense, and $O(1/\sum_{i=0}^{k}a_{i+1})$ for $\mathtt{Gap}(\cdot)$ in the ergodic sense. Second, to enhance the convergence rate of the newly-proposed PPA, we introduce an accelerated variant called the Contracting PPA. By utilizing a resolvent index of $T$ bounded by $O(a_{k+1})$ ($a_{k+1}\geq \varepsilon>0$), we establish a convergence rate of $O(1/\sum_{i=0}^{k}a_{i+1})$ for both $\|T_{\lambda}(\cdot)\|$ and $\mathtt {Gap}(\cdot)$, considering the non-ergodic sense. Third, to mitigate the limitation that the Contracting PPA lacks a convergence guarantee, we propose two additional versions of the algorithm. These novel approaches not only ensure guaranteed convergence but also provide sublinear and linear convergence rates for both $\|T_{\lambda}(\cdot)\|$ and $\mathtt {Gap}(\cdot)$, respectively, in the non-ergodic sense.