lect4 - ISE 536Fall03: Linear Programming and Extensions...

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Unformatted text preview: ISE 536Fall03: Linear Programming and Extensions September 3, 2003 Lecture 4: Geometry, BFS in Standard Form Lecturer: Fernando Ord onez 1 BFS vertex extreme point Let us recall the definitions x is an extreme point of a polyhedron P if it cant 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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lect4 - ISE 536Fall03: Linear Programming and Extensions...

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