geometry_lp - IEOR 162 NOTES SPRING 2007 1. The geometry of...

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Unformatted text preview: IEOR 162 NOTES SPRING 2007 1. The geometry of linear programming Let us consider the geometry of a linear program in n-dimensional space. Definition 1. H = { x R n : ax b } , where a is an n-dimensional row vector, is called a half space . Definition 2. P = { x R n : a i x b i ,i = 1 ,...,m } = { x R n : Ax b } is called a polyhedron , where A is an m n matrix, b is a m-dimensional vector. Definition 3. Extreme point of a polyhedron P is a point in P that can not be written as a strict convex combination of two points in P , i.e., x P is an extreme point of P if and only if x 6 = x 1 + (1- ) x 2 for any x 1 ,x 2 P , x 1 6 = x 2 and 0 < < 1. Definition 4. For x R n if a x = b , we say that ax b is active or binding at point x . Definition 5. x is a basic solution of P if there are n linear independent constraints ( a i ) of P that are active at x . x is called a basic feasible solution (bfs) of P if x is a basic solution and feasible....
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