# lect11 - ISE 536Fall03 Linear Programming and Extensions...

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ISE 536–Fall03: Linear Programming and Extensions October 6, 2003 Lecture 11: Duality, Introduction Lecturer: Fernando Ord´ o˜nez 1 Definition Here the rows of the matrix A are a t i , for i = 1 , . . . , m , and the columns are A j , j = 1 , . . . , n . For any linear programming problem: ( P ) min c t x s . t . a t i x = b i i M 1 a t i x b i i M 2 a t i x b i i M 3 x j 0 j N 1 x j 0 j N 2 x j unbounded j N 3 there exists a closely related LP problem called the dual: ( D ) max y t b s . t . y t A j c j j N 1 y t A j c j j N 2 y t A j = c j j N 3 y i unbounded i M 1 y i 0 i M 2 y i 0 i M 3 1

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For example, construct the dual of the problem ( P ) min 2 x 1 + x 2 s . t . - x 1 + 3 x 2 = 5 2 x 1 + 7 x 2 3 x 1 1 x 1 0 x 2 unrestricted 2 Relation between ( P ) and ( D ) In matrix notation we have that ( P ) min c t x s . t . Ax = b x 0 then ( D ) max y t b s . t . y t A c t This dual problem is sometimes expressed in the more symmetric form ( D ) max b t y s . t . A t y + s = c s 0 Theorem The dual of the dual is the primal.
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