simplexI

# simplexI - OR320/520 Prof Bland The Simplex Method Parts I...

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OR320/520 9/13/07 Prof. Bland The Simplex Method: Parts I and II Part I: Standard Form Systems and Ordered Bases Let A be an m × n matrix and let b be an m × 1 vector. We will be interested in linear programming problems in which the constraints take the form Ax = b, x 0. We observed in the last class that the set S = { x R I n : Ax = b, x 0 } of feasible solutions to such l.p. problems is a polyhedron. This polyhedron may not be bounded (for example if m = 0), but if it is nonempty, it has at least one extreme point. (We will give an intuitive argument in class.) For a little while we are going to ignore the nonnegativity constraints and focus on the linear system Ax = b . For each index 1 j n let A j denote the jth column of A . Now suppose that for the list B = ( B 1 , ..., B m ) of indices in { 1 , ..., n } the m × m matrix A B = [ A B 1 , ..., A B m ] is nonsingular. Then we say that B is basic for A , and that B is an ordered basis correspond- ing to the linear system ¯ Ax = ¯ b , where ( ¯ A, ¯ b ) := A - 1 B ( A, b ) The variables x B 1 , ··· , x B m are basic variables , and the other variables x j , for j not in B , are called nonbasic . It is convenient to denote by N the set of indices of the nonbasic variables. For example let

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## This note was uploaded on 03/30/2008 for the course ORIE 320 taught by Professor Bland during the Fall '07 term at Cornell.

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simplexI - OR320/520 Prof Bland The Simplex Method Parts I...

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