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L7_10 - ESI 6314 Deterministic Methods in Operations...

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1/34 ESI 6314 Deterministic Methods in Operations Research Lecture Notes 8 University of Florida Department of Industrial and Systems Engineering / REEF

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2/34 Duality theory max z = 5 x 1 + 5 x 2 + 3 x 3 s . t . x 1 + 3 x 2 + x 3 3 x 1 + 3 x 3 2 2 x 1 - x 2 + 2 x 3 4 2 x 1 + 3 x 2 - x 3 2 x 1 , x 2 , x 3 0 How can we derive an upper bound estimate for the optimal value z * of z ? For example, if we multiply the first constraint by 5, we have z = 5 x 1 + 5 x 2 + 3 x 3 5 x 1 + 15 x 2 + 5 x 3 15 for any x 1 , x 2 , x 3 0. So, z * 15.
3/34 Duality theory More generally, to get an upper bound on z , we can multiply the i -th constraint by y i 0 and then add the resulting inequalities together: max z = 5 x 1 + 5 x 2 + 3 x 3 x 1 + 3 x 2 + x 3 3 × y 1 x 1 + 3 x 3 2 × y 2 2 x 1 - x 2 + 2 x 3 4 × y 3 2 x 1 + 3 x 2 - x 3 2 × y 4 We obtain: ( y 1 + y 2 + 2 y 3 + 2 y 4 ) x 1 +(3 y 1 - y 3 + 3 y 4 ) x 2 +( y 1 + 3 y 2 + 2 y 3 - y 4 ) x 3 3 y 1 + 2 y 2 + 4 y 3 + 2 y 4

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4/34 Duality theory We have z = 5 x 1 + 5 x 2 + 3 x 3 and ( y 1 + y 2 + 2 y 3 + 2 y 4 ) x 1 +(3 y 1 - y 3 + 3 y 4 ) x 2 +( y 1 + 3 y 2 + 2 y 3 - y 4 ) x 3 3 y 1 + 2 y 2 + 4 y 3 + 2 y 4 If we make sure that y 1 + y 2 + 2 y 3 + 2 y 4 5 3 y 1 - y 3 + 3 y 4 5 y 1 + 3 y 2 + 2 y 3 - y 4 3 then z = 5 x 1 + 5 x 2 + 3 x 3 3 y 1 + 2 y 2 + 4 y 3 + 2 y 4
5/34 Duality theory In order to have as tight upper bound on z as possible, we want to minimize 3 y 1 + 2 y 2 + 4 y 3 + 2 y 4 . We obtain the dual LP : min w = 3 y 1 + 2 y 2 + 4 y 3 + 2 y 4 s . t . y 1 + y 2 + 2 y 3 + 2 y 4 5 3 y 1 - y 3 + 3 y 4 5 y 1 + 3 y 2 + 2 y 3 - y 4 3 y 1 , y 2 , y 3 , y 4 0 The original LP is called the primal LP

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6/34 Duality theory Primal LP: max z = 5 x 1 + 5 x 2 + 3 x 3 s . t . x 1 + 3 x 2 + x 3 3 x 1 + 3 x 3 2 2 x 1 - x 2 + 2 x 3 4 2 x 1 + 3 x 2 - x 3 2 x 1 , x 2 , x 3 0 Dual LP: min w = 3 y 1 + 2 y 2 + 4 y 3 + 2 y 4 s . t . y 1 + y 2 + 2 y 3 + 2 y 4 5 3 y 1 - y 3 + 3 y 4 5 y 1 + 3 y 2 + 2 y 3 - y 4 3 y 1 , y 2 , y 3 , y 4 0
7/34 Duality theory In general, if the primal LP is max z = n j =1 c j x j s . t . n j =1 a ij x j b i , i = 1 , . . . , m x 1 , . . . , x n 0 then the dual LP is given by min w = m i =1 b i y i s . t . m i =1 a ij y i c j , j = 1 , . . . , n y 1 , . . . , y m 0 x j are called primal variables y i are called dual variables

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8/34 Duality theory z = n X j =1 c j x j n X j =1 m X i =1 a ij y i ! x j = m X i =1 n X j =1 a ij x j y i m X i =1 b i y i = w Hence, for any feasible solution x to the primal LP and any feasible solution y to the dual LP we have z = n X j =1 c j x j m X i =1 b i y i = w [ weak duality ] Therefore, if we find x * (feasible for the primal LP) and y * (feasible for the dual LP) such that n X j =1 c j x * j = m X i =1 b i y * i then x * is an optimal solution of the primal problem and y * is an optimal solution of the dual problem!
9/34 Duality theory Primal LP : max z = n j =1 c j x j s . t . n j =1 a ij x j b i , i = 1 , . . . , m x 1 , . . . , x n 0 Dual LP : min w = m i =1 b i y i s . t .

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