05 - Industrial Engineering 634 Integer Programming Fall...

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Industrial Engineering 634 Fall 2011 Integer Programming Lecturer: Nelson Uhan Scribe: Chong Hyun Park August 31, 2011 Lecture 5. LP Duality Let us recall Corollary F from Lecture 3. Corollary 1 (Corollary F in Lecture 3) . All minimal faces of { x R n : Ax b } have dimension n - rank ( A ) . For this lecture, we have A R m × n ,b R m ,c R n . We define P = { x R n : Ax b } , D = { y R m : yA = c, y 0 } , z P = max { cx : x P } , (P) z D = min { yb : y D } . (D) We say that (P) is the primal LP, and (D) is the dual of (P). Proposition 1. The dual of the dual is the primal. Proof. Let (P) be given. (D) can be written equivalently as: - maximize - yb subject to A T y c - A T y ≥ - c, - y 0 . So the dual of (D) is: - minimize cz - cz 0 (P*) subject to Az - Az 0 - w = - b, z,z 0 ,w 0 . Substitute x = z 0 - z and regard w as slack variables. Then, we have that (P*) is equivalent to (P). Theorem 2
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05 - Industrial Engineering 634 Integer Programming Fall...

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