▁Fr isch ▁in ▁addition ▁to ▁Kant or ov ich , ▁Hur w icz , ▁Ko op mans , ▁Ar row , ▁and ▁Samuel son . ▁ ▁Linear ▁optimization ▁ ▁Linear ▁programming ▁was ▁developed ▁to ▁aid ▁the ▁allocation ▁of ▁resources ▁in ▁fir ms ▁and ▁in ▁indust ries ▁during ▁the ▁ 1 9 3 0 s ▁in ▁Russia ▁and ▁during ▁the ▁ 1 9 4 0 s ▁in ▁the ▁United ▁States . ▁During ▁the ▁Berlin ▁air l ift ▁( 1 9 4 8 ), ▁linear ▁programming ▁was ▁used ▁to ▁plan ▁the ▁ship ment ▁of ▁supplies ▁to ▁prevent ▁Berlin ▁from ▁star ving ▁after ▁the ▁Soviet ▁ ▁block ade . ▁ ▁Non linear ▁programming ▁ ▁Ext ensions ▁to ▁non linear ▁optimization ▁with ▁inequality ▁constraints ▁were ▁achieved ▁in ▁ 1 9 5 1 ▁by ▁Albert ▁W . ▁T ucker ▁and ▁Harold ▁Ku hn , ▁who ▁considered ▁the ▁non linear ▁optimization ▁problem : ▁Min im ize ▁() ▁subject ▁to ▁i () ▁ ≤ ▁ 0 ▁and ▁j () ▁= ▁ 0 ▁where ▁( .) ▁is ▁the ▁function ▁to ▁be ▁minim ized ▁i ( .) ▁( ▁= ▁ 1 , ▁... , ▁) ▁are ▁the ▁functions ▁of ▁the ▁ ▁inequality ▁constraints ▁j ( .) ▁( ▁= ▁ 1 , ▁... , ▁) ▁are ▁the ▁functions ▁of ▁the ▁ ▁equality ▁constraints . ▁ ▁In ▁allowing ▁inequality ▁constraints , ▁the ▁Ku hn – T ucker ▁approach ▁generalized ▁the ▁classic ▁method ▁of ▁Lag range ▁multip liers , ▁which ▁( until ▁then ) ▁had ▁ ▁allowed ▁only ▁equality ▁constraints . ▁ ▁The ▁Ku hn