Lecture14 - Lecture 14: Linear Programming Duality...

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Lecture 14: Linear Programming Duality September 27, 2010

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The Dual Linear Program Given an LP in the standard form, max(min) z = c 1 x 1 + c 2 x 2 + ··· + c n x n s.t. a 11 x 1 + a 12 x 2 + ··· + a 1 n x n = b 1 ······ y 1 a 21 x 1 + a 22 x 2 + ··· + a 2 n x n = b 2 ······ y 2 . . . . . . . . . . . . = . . . ······ . . . a m 1 x 1 + a m 2 x 2 + ··· + a mn x n = b m ······ y m x j 0 j = 1 , ··· ,n introducing dual variables for all the equality constraints, the dual problem is given by: min(max) w = b 1 y 1 + b 2 y 2 + ··· + b m y m s.t. a 11 y 1 + a 21 y 2 + ··· + a m 1 y m ( ) c 1 a 12 y 1 + a 22 y 2 + ··· + a m 2 y m ( ) c 2 . . . . . . . . . . . . ( ) . . . a 1 n y 1 + a 2 n y 2 + ··· + a mn y m ( ) c n y i unristricted i = 1 , ··· ,m We call the original problem the primal problem. 2
The Dual Linear Program: Examples Primal: max z = 5 x 1 + 12 x 2 + 4 x 3 s.t. x 1 + 2 x 2 + x 3 10 2 x 1 - x 2 + 3 x 3 = 8 x 1 , x 2 , x 3 0 Add slack variables:

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This note was uploaded on 09/27/2010 for the course GE 330 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Lecture14 - Lecture 14: Linear Programming Duality...

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