section5 - 5 LP Geometry We now briey turn to a discussion...

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5 LP Geometry We now briefly turn to a discussion of LP geometry extending the geometric ideas developed in Section 1 for 2 dimensional LPs to n dimensions. In this regard, the key geometric idea is the notion of a hyperplane. Definition 5.1 A hyperplane in R n is any set of the form H ( a, β ) = { x : a T x = β } where a R n , β R , and a negationslash = 0 . We have the following important fact whose proof we leave as an exercise for the reader. Fact 5.2 H R n is a hyperplane if and only if the set H - x 0 = { x - x 0 : x H } where x 0 H is a subspace of R n of dimension ( n - 1) . Every hyperplane H ( a, β ) generates two closed half spaces: H + ( a, β ) = { x R n : a T x β } and H - ( a, β ) = { x R n : a T x β } . Note that the constraint region for a linear program is the intersection of finitely many closed half spaces: setting H j = { x : e T j x 0 } for j = 1 , . . . , n and H n + i = { x : n summationdisplay j =1 a ij x j b i } for i = 1 , . . . , m we have { x : Ax b, 0 x } = n + m intersectiondisplay i =1 H i . Any set that can be represented in this way is called a convex polyhedron . Definition 5.3 Any subset of R n that can be represented as the intersection of finitely many closed half spaces is called a convex polyhedron. Therefore, a linear programming is simply the problem of either maximizing or minimizing a linear function over a convex polyhedron. We now develop some of the underlying geometry of convex polyhedra. 56
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Fact 5.4 Given any two points in R n , say x and y , the line segment connecting them is given by [ x, y ] = { (1 - λ ) x + λy : 0 λ 1 } . Definition 5.5 A subset C R n is said to be convex if [ x, y ] C whenever x, y C . Fact 5.6 A convex polyhedron is a convex set. We now consider the notion of vertex, or corner point, for convex polyhedra in R 2 . For this, consider the polyhedron C R 2 defined by the constraints c 1 : - x 1 - x 2 ≤ - 2 (5.6) c 2 : 3 x 1 - 4 x 2 0 c 3 : - x 1 + 3 x 2 6 . v 2 c 3 c 1 c 2 x 1 x 2 1 1 2 2 3 3 4 4 5 5 v 3 v 1 C The vertices are v 1 = ( 8 7 , 6 7 ) , v 2 = (0 , 2), and v 3 = ( 24 5 , 18 5 ) . One of our goals in this section is to discover an intrinsic geometric property of these vertices that generalizes to n dimensions and simultaneously captures our intuitive notion of what a vertex is. For this we examine our notion of convexity which is based on line segments. Is there a way to use line segments to make precise our notion of vertex? 57
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Consider any of the vertices in the polyhedron C defined by (5.7). Note that any line segment in C that contains one of these vertices must have the vertex as one of its end points. Vertices are the only points that have this property. In addition, this property easily generalizes to convex polyhedra in R n . This is the rigorous mathematical formulation for our notion of vertex that we seek. It is simple, has intuitive appeal, and yields the correct objects in dimensions 2 and 3.
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