– T ucker ▁approach ▁inspired ▁further ▁research ▁on ▁L agr ang ian ▁du ality , ▁including ▁the ▁treatment ▁of ▁inequality ▁constraints . ▁The ▁du ality ▁theory ▁of ▁non linear ▁programming ▁is ▁particularly ▁satisf actory ▁when ▁applied ▁to ▁convex ▁minim ization ▁problems , ▁which ▁enjoy ▁the ▁convex - anal yt ic ▁du ality ▁theory ▁of ▁Fen ch el ▁and ▁Rock af ell ar ; ▁this ▁convex ▁du ality ▁is ▁particularly ▁strong ▁for ▁poly hed ral ▁convex ▁functions , ▁such ▁as ▁those ▁ar ising ▁in ▁linear ▁programming . ▁L agr ang ian ▁du ality ▁and ▁convex ▁analysis ▁are ▁used ▁daily ▁in ▁operations ▁research , ▁in ▁the ▁sched uling ▁of ▁power ▁plants , ▁the ▁planning ▁of ▁production ▁sched ules ▁for ▁fact ories , ▁and ▁the ▁routing ▁of ▁air lines ▁( routes , ▁fl ights , ▁plan es , ▁cre ws ). ▁ ▁Vari ational ▁calculus ▁and ▁optimal ▁control ▁ ▁Econom ic ▁dynamics ▁allows ▁for ▁changes ▁in ▁economic ▁variables ▁over ▁time , ▁including ▁in ▁dynamic ▁systems . ▁ ▁The ▁problem ▁of ▁finding ▁optimal ▁functions ▁for ▁such ▁changes ▁is ▁studied ▁in ▁vari ational ▁calculus ▁and ▁in ▁optimal ▁control ▁theory . ▁ ▁Before ▁the ▁Second ▁World ▁War , ▁Frank ▁Ram sey ▁and ▁Harold ▁Hot elling ▁used ▁the ▁calculus ▁of ▁variations ▁to ▁that ▁end . ▁ ▁Following ▁Richard ▁Bell man ' s ▁work ▁on ▁dynamic ▁programming ▁and ▁the ▁ 1 9 6 2 ▁English ▁translation ▁of ▁L . ▁Pon try agin ▁et ▁al .' s ▁earlier ▁work , ▁optimal ▁control ▁theory ▁was ▁used ▁more ▁extens ively ▁in ▁econom ics ▁in