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# lect5 - ISE 536Fall03 Linear Programming and Extensions...

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ISE 536–Fall03: Linear Programming and Extensions September 15, 2003 Lecture 5: Geometry, Optimality of BFS Lecturer: Fernando Ord´ o˜nez 1 Existence of a BFS Theorem 1. P = { x n | Ax b } = . P has a BFS iff P does not contain a line Proof: pg. 63-64. Check that this result implies the following Bounded polyhedra must have a BFS A polyhedra in standard form must have a BFS 2 Optimality of a BFS Theorem 2. If min { c t x | Ax = b, x 0 } has an optimal solution then there is an optimal BFS proof: Consider Q = { x | c t x = z * , Ax = b, x 0 } 1

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3 Unbounded Polyhedra and Extreme Rays Consider P = { x | Ax = b, x 0 } an unbounded polyhedra. Therefore there is a direction
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lect5 - ISE 536Fall03 Linear Programming and Extensions...

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