# lec10 - 15.053 Duality 3 There are concepts much more...

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15.053 March 13, 2007 Duality 3 There are concepts much more difficult to grasp than duality in linear programming. -- Jim Orlin The concept [of nonduality], often described in English as "nondualism," is extremely hard for the mind to grasp or visualize, since the mind engages constantly in the making of distinctions and nondualism represents the rejection or transcendence of all distinctions. from www.nonduality.com 1

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Overview z Rules for creating a dual linear program z Complementary slackness conditions z The dual simplex algorithm 2
Rules for creating a dual linear program The Primal LP Dual Maximize z = 1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 Variable subject to 8 x 1 + 9 x 2 + 10 x 3 + 11 x 4 ? 5 y 1 12 x 1 + 13 x 2 + 14 x 3 + 15 x 4 ? 6 y 2 16 x 1 + 17 x 2 + 18 x 3 + 19 x 4 ? 7 y 3 x 1 ?x 2 ? x 3 4 ? The Dual LP Minimize v = 5 y 1 + 6 y 2 + 7 y 3 subject to 8 y 1 + 12 y 2 + 16 y 3 ? 1 9 y 1 + 13 y 2 + 17 y 3 ? 2 10 y 1 + 14 y 2 + 18 y 3 ? 3 11 y 1 + 15 y 2 + 19 y 3 ? 4 y 1 ?y 2 ? y 3 ? 3

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= 4 The SOB approach sensible, odd, and bizarre The Primal LP Max z = 1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 s.t. 8 x 1 + 9 x 2 + 10 x 3 + 11 x 4 5 12 x 1 + 13 x 2 + 14 x 3 + 15 x 4 6 16 x 1 + 17 x 2 + 18 x 3 + 19 x 4 7 x 1 0, x 2 0, x 3 uis, x 4 0? S O B S S O B uis. unconstrained in sign =
= 5 The SOB approach sensible, odd, and bizarre The Dual LP Min v = 5 y 1 + 6 y 2 + 7 y 3 s.t 8 y 1 + 12 y 2 + 16 y 3 1 9 y 1 + 13 y 2 + 17 y 3 2 10 y 1 + 14 y 2 + 18 y 3 3 11 y 1 + 15 y 2 + 19 y 3 4 y 1 0, y 2 uis y 3 0 S S O B S O B =

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More Rules Primal max Dual min i-th constraint i-th variable 0 S i-th constraint = i-th variable free O i-th constraint i-th variable 0 B i-th variable 0 i-th constraint S i-th variable free i-th constraint = O i-th variable 0 i-th constraint B S: sensible O: odd The SOB technique is due to Art B: bizarre Benjamin 6
= 0, 0 = 0, free, 0 7 The Primal LP Maximize z = 1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 subject to 8 x 1 + 9 x 2 + 10 x 3 + 11 x 4 5 12 x 1 + 13 x 2 + 14 x 3 + 15 x 4 6 16 x 1 + 17 x 2 + 18 x 3 + 19 x 4 7 x 1 x 2 0, x 3 free, x 4 The Dual LP Minimize v = 5 y 1 + 6 y 2 + 7 y 3 subject to 8 y 1 + 12 y 2 + 16 y 3 1 9 y 1 + 13 y 2 + 17 y 3 2 10 y 1 + 14 y 2 + 18 y 3 3 11 y 1 + 15 y 2 + 19 y 3 4 y 1 y 2 y 3 a b c d e f g d e f g a b c

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0, 0 8 Shadow Price Method The Primal LP Maximize z = 1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 subject to 8 x 1 + 9 x 2 + 10 x 3 + 11 x 4 5 12 x 1 + 13 x 2 + 14 x 3 + 15 x 4 = 6 16 x 1 + 17 x 2 + 18 x 3 + 19 x 4 7 x 1 x 2 0, x 3 free, x 4 d e f g RHS of constraint (1) increases from 5 to 5.1 implies Primal LP is less constrained implies z increases or stays the same implies y 1 0.
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lec10 - 15.053 Duality 3 There are concepts much more...

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