Abstract
Recently, a worst-case <svg style="vertical-align:-2.3205pt;width:45.900002px;" id="M1" height="15.0875" version="1.1" viewBox="0 0 45.900002 15.0875" width="45.900002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D442" d="M745 361q0 -166 -116 -272t-291 -106q-136 0 -225.5 83t-89.5 218q0 161 115.5 272t288.5 111q137 0 227.5 -83t90.5 -223zM643 359q0 126 -56 199.5t-167 73.5q-130 0 -213.5 -104t-83.5 -246q0 -117 58 -190.5t165 -73.5q91 0 160.5 52t103 128.5t33.5 160.5z" /></g><g transform="matrix(.017,-0,0,-.017,13.118,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,18.999,12.138)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,27.159,12.138)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.017,-0,0,-.017,34.162,12.138)"><path id="x1D461" d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" /></g><g transform="matrix(.017,-0,0,-.017,39.959,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg> convergence rate was established for the Douglas-Rachford alternating direction method of multipliers (ADMM) in an ergodic sense. The relaxed proximal point algorithm (PPA) is a generalization of the original PPA which includes the Douglas-Rachford ADMM as a special case. In this paper, we provide a simple proof for the same convergence rate of the relaxed PPA in both ergodic and nonergodic senses.
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