A note on the inertial proximal point method - Zhenguo Mu; Yang Peng (Statistics, Optimization & Information Computing, 2015)
The proximal point method (PPM) for solving maximal monotone operator inclusion problem is a highly powerful tool for algorithm design, analysis and interpretation. To accelerate convergence of the PPM,   inertial PPM (iPPM) was proposed in the literature. In this  note, we point out that some of the attractive properties of the PPM, e.g., the generated sequence is contractive with the set of solutions, do not hold anymore for iPPM. To partially inherit the advantages of the PPM and meanwhile incorporate inertial extrapolation steps, we propose   an iPPM with alternating inertial steps. Our analyses show that the even subsequence generated by the proposed iPPM is contractive with the set of solutions. Moreover, we establish global convergence result under much relaxed conditions on the inertial extrapolation stepsizes, e.g., monotonicity is no longer needed and the stepsizes are significantly enlarged compared to existing methods. Furthermore, we establish certain   $o(1/k)$  convergence rate results, where $k$ denotes the iteration counter. These features are new to inertial type PPMs.