▁for ▁y ▁with ▁x ▁fixed . ▁R ather ▁than ▁iterate ▁until ▁convergence ▁( like ▁the ▁Jacob i ▁method ), ▁the ▁algorithm ▁proceed s ▁directly ▁to ▁updating ▁the ▁dual ▁variable ▁and ▁then ▁repeating ▁the ▁process . ▁ ▁This ▁is ▁not ▁equivalent ▁to ▁the ▁exact ▁minim ization , ▁but ▁surprising ly , ▁it ▁can ▁still ▁be ▁shown ▁that ▁this ▁method ▁converges ▁to ▁the ▁right ▁answer ▁( under ▁some ▁assumptions ). ▁Because ▁of ▁this ▁approximation , ▁the ▁algorithm ▁is ▁distinct ▁from ▁the ▁pure ▁augment ed ▁L agr ang ian ▁method . ▁ ▁The ▁AD MM ▁can ▁be ▁viewed ▁as ▁an ▁application ▁of ▁the ▁Douglas - R ach ford ▁splitting ▁algorithm , ▁and ▁the ▁Douglas - R ach ford ▁algorithm ▁is ▁in ▁turn ▁an ▁instance ▁of ▁the ▁Pro xim al ▁point ▁algorithm ; ▁details ▁can ▁be ▁found ▁here . ▁ ▁There ▁are ▁several ▁modern ▁software ▁packages ▁that ▁solve ▁Bas is ▁purs uit ▁and ▁variants ▁and ▁use ▁the ▁AD MM ; ▁such ▁packages ▁include ▁Y ALL 1 ▁( 2 0 0 9 ), ▁Sp a R SA ▁( 2 0 0 9 ) ▁and ▁S AL SA ▁( 2 0 0 9 ). ▁There ▁are ▁also ▁packages ▁that ▁use ▁the ▁AD MM ▁to ▁solve ▁more ▁general ▁problems , ▁some ▁of ▁which ▁can ▁explo it ▁multiple ▁computing ▁cores ▁ ▁SN AP V X ▁( 2 0 1 5 ), ▁par AD MM ▁( 2 0 1 6 ). ▁ ▁Sto ch astic ▁optimization ▁ ▁Sto ch astic ▁optimization ▁cons iders ▁the ▁problem ▁of ▁minim izing ▁a ▁loss ▁function ▁with