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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online