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04 - Industrial Engineering 634 Integer Programming Fall...

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Industrial Engineering 634 Fall 2011 Integer Programming Lecturer: Nelson Uhan Scribe: Isaac Tetzloff August 29, 2011 Lecture 4. The Simplex Method and LP Duality Let J be a set of row indices. With J , we let A J be the submatrix of A with rows in J , and we let b J be the subvector of b with components in J . Algorithm 1 (The Simplex Algorithm) . The Simplex Algorithm for solving a linear program (LP) takes in the following as input: A R m × n , b R m , c R n , and a vertex x of P , { x R n : Ax b } . After the Simplex Algorithm terminates the following output is given: a vertex x of P attaining max { cx : x P } or a vector w R n with Aw 0 and cw > 0 (i.e., the LP is unbounded). The process for the Simplex Algorithm is given by the following steps: 1. Find an initial basis by choosing a set of n row indices J such that A J is nonsingular and A J x = b J . 2. Compute the reduced costs by computing cA - 1 J and add zeros in order to obtain a vector y with c = yA , and all entries of y outside of J are 0. If y
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