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# lect17 - Lecture 17 Linear Programming II Duality Dual LP...

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Lecture 17 Linear Programming II. Duality

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Dual LP . 0 s.t. max = = x b Ax cx z s.t. min c A y b y z T T = Primal Dual
Lemma cx Ax y b y b y cx y x T T T = . , point feasible dual any and point feasible primal any For Proof .

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. * * Moreover, ly. respective , * and * say solutions, optimal have both they then solution, feasible have dual the and primal both the If solution. feasible no has dual the iff value maximum has primal The . value minimum has dual the iff solution feasible no has primal The b y cx y x T = + - Theorem
Condition. Optimality the of proof the Recall Proof.

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Optimality Condition condition. acy nondegener under 0 if only and if optimal is Moreover, }. | { where 0 and with solution feasible basic a with associated is basis feasible Each 1 1 - = = = - - I I I I I I I A A c c x I j j I x b A x x I
Degeneracy Condition . 0 , basis feasible every For 1 - b A I I

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Sufficiency . 0 at maximum the reaches , 0 and 0 Since ) ( 1 1 1 = - - + = + = - = = + = - - - I I I I I I I I I I I I I I I I I I I I I I I I x cx x A A c c x A A c c b c x c x c cx x A A b x b x A x A b Ax
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