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Unformatted text preview: ISE 536–Fall03: Linear Programming and Extensions September 3, 2003 Lecture 4: Geometry, BFS in Standard Form Lecturer: Fernando Ord´ o˜nez 1 BFS ⇔ vertex ⇔ extreme point Let us recall the definitions • x is an extreme point of a polyhedron P if it can’t be expressed as a convex combi nation of two other points in P . • x is a vertex of a polyhedron P if it is the unique optimal solution of min { c t y  y ∈ P } for some objective function c • x is a BFS if { a i } i ∈ I spans < n , where I is the set of tight constraints. Proof: 1) vertex ⇒ extreme point 2) extreme point ⇒ BFS 3) BFS ⇒ vertex 1 2 BFS in standard form Consider P = { x ∈ < n  Ax = b,x ≥ } , where A ∈ < m × n with m < n and rank( A ) = m . A BFS needs to complete the m tight constraints from Ax = b with n m constraints x j = 0 that are linearly independent with the rows of A . Partition the variables x t = ( x 1 ,...,x m ,x m +1 ,...,x n ) and the constraint matrix A = [ BN ]: Definition for standard form:...
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This note was uploaded on 02/13/2012 for the course ISE 536 taught by Professor Yy during the Spring '05 term at South Carolina.
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