Lecture 9D

Lecture 9D - Simplex Algorithm: Column Generation a 5 , a 6...

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EE103 Lec 9D (SEJ) 1 Simplex Algorithm: Column Generation () 54 1 ,0 0,40 1 z aa B ⎡⎤ ⎣⎦ = ⎢ 24 2 20,40 1 z B = ⎢⎥ 14 3 12,40 1 z B = 56 0 0,0 1 z B = ⎢ EE103 Lec 9D (SEJ) 2 Column Generation: Implementation 1 11 1 1 1 1 0 00 1 z new tt t t t t NN B B B B B Nb B N B b BI B TT cc c B N B b The basic structure at every iteratio cB n −− ⎛⎞ == = ⎜⎟ ⎝⎠ 1 1 1 , BN tj jj B t B xB b x jc c c B a zcBb ∈ℵ = = variables where is the index set of non basic 0 0 1 t t N B B T c c = 1 1 1 0 0 , 1 1 zz t t B B B B BB c 0 0 1 T b A original data c
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EE103 Lec 9D (SEJ) 3 Column Generation: Implementation ? 1 max :0 Tj jj B Assume the LP is a problem jc c c B a ∈ℵ = 1 def TT B yc B = ? c y a ℵ= B T B yB c or By c = = Assume wehave PB LU = I 0 ? # # 0 0 ? 0 ? 1 ? B x z 1 ,, m aa " EE103 Lec 9D (SEJ) 4 Column Generation: Implementation B T B yB c o r = = Assume wehave PB LU = T B ULP = T B B = B U L Py c = B T T B T uP y ULu c let v L u Uv c v L uv u = = = = = T y Pu =
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EE103 Lec 9D (SEJ) 5 Column Generation: Implementation 1 , 0 () : Tj jj B z Therefore if j ccc B a B or B is an optimal basis ∀∈ℵ =− 1 1 : ,0 BN TT B Optimal xB b x zc B b y b == , 0 ' 0 k However assume ccy a and we ve selected k to enter basis c
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This note was uploaded on 03/30/2009 for the course EE 103 taught by Professor Vandenberghe,lieven during the Winter '08 term at UCLA.

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Lecture 9D - Simplex Algorithm: Column Generation a 5 , a 6...

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