IOE510+Lecture+02+24+10+Wed

IOE510+Lecture+02+24+10+Wed - IOE 510 Linear Programming I...

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1 IOE 510 Linear Programming I Wednesday February 24, 2010 Professor Amy Cohn
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2 Announcements Re-grade policy No problem set 7
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3 Re-cap from Last Time Finalizing simplex method “Big-M” method for generating initial BFS Some observations about degeneracy
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4 Generating An Initial BFS Consider an LP of the form Min cx St Ax = b x > 0 WLOG assume b i > 0 for all i
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5 A New LP (The Big-M Method) Construct a new LP: Min cx + Mey St Ax + Iy = b x > 0 y > 0 where y R m , e is a row vector in R m containing all 1’s, and M is a “very large number”
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6 Example Min 2x 1 + 3x 2 + 4x 3 St x 1 – x 2 = 7 x 1 + x 2 – x 3 = 3 x 1 , x 2 , x 3 > 0 Min 2x 1 + 3x 2 + 4x 3 + My 1 + My 2 St x 1 – x 2 + y 1 = 7 x 1 + x 2 – x 3 + y 2 = 3 x 1 , x 2 , x 3 , y 1 , y 2 > 0 What is the initial basis? What is the first pivot?
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7 Solving the New Problem Claim: (x = 0, y = b) is a BFS to this new problem Given this initial BFS, we can solve the new problem using the simplex algorithm Let (x*, y*) be an optimal solution to the new problem
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IOE510+Lecture+02+24+10+Wed - IOE 510 Linear Programming I...

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