▁variable ▁ ▁is ▁an ▁estimate ▁of ▁the ▁Lag range ▁multip lier , ▁and ▁the ▁accuracy ▁of ▁this ▁estimate ▁impro ves ▁at ▁every ▁step . ▁ ▁The ▁major ▁advantage ▁of ▁the ▁method ▁is ▁that ▁unlike ▁the ▁penalty ▁method , ▁it ▁is ▁not ▁necessary ▁to ▁take ▁ ▁in ▁order ▁to ▁solve ▁the ▁original ▁const rained ▁problem . ▁ ▁Instead , ▁because ▁of ▁the ▁presence ▁of ▁the ▁Lag range ▁multip lier ▁term , ▁ ▁can ▁stay ▁much ▁smaller , ▁thus ▁avoid ing ▁ill - condition ing . ▁ ▁The ▁method ▁can ▁be ▁extended ▁to ▁handle ▁inequality ▁constraints . ▁ ▁For ▁a ▁discussion ▁of ▁practical ▁improvements , ▁see . ▁ ▁Altern ating ▁direction ▁method ▁of ▁multip liers ▁ ▁The ▁altern ating ▁direction ▁method ▁of ▁multip liers ▁( AD MM ) ▁is ▁a ▁variant ▁of ▁the ▁augment ed ▁L agr ang ian ▁scheme ▁that ▁uses ▁partial ▁updates ▁for ▁the ▁dual ▁variables . ▁This ▁method ▁is ▁often ▁applied ▁to ▁solve ▁problems ▁such ▁as ▁ ▁This ▁is ▁equivalent ▁to ▁the ▁const rained ▁problem ▁ ▁Though ▁this ▁change ▁may ▁seem ▁trivial , ▁the ▁problem ▁can ▁now ▁be ▁attacked ▁using ▁methods ▁of ▁const rained ▁optimization ▁( in ▁particular , ▁the ▁augment ed ▁L agr ang ian ▁method ), ▁and ▁the ▁objective ▁function ▁is ▁separ able ▁in ▁x ▁and ▁y . ▁ ▁The ▁dual ▁update ▁requires ▁solving ▁a ▁proxim ity ▁function ▁in ▁x ▁and ▁y ▁at ▁the ▁same ▁time ; ▁the ▁AD MM ▁technique ▁allows ▁this ▁problem ▁to ▁be ▁solved ▁approximately ▁by ▁first ▁solving ▁for ▁x ▁with ▁y ▁fixed , ▁and ▁then ▁solving