Geometry of Solutions

Geometry of Solutions - Economics 146 Fall 2007 Linear...

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Economics 146 – Fall, 2007 Linear Programming – Part IIIB Last revision: 10/26/07. Geometry of Solutions. Let S be the set of feasible solutions for an LP in standard form. We can classify three possible cases for S . We first establish some terminology. Consider the feasible set given by Ax b (1) and x 0 (2) Another way to write the conditions is a 11 x 1 + a 12 x 2 + · · · + a 1 n x n b 1 (3) a 21 x 1 + a 22 x 2 + · · · + a 1 n x n b 2 (4) . . . . . . (5) a m 1 x 1 + a m 2 x 2 + · · · + a mn x n b m (6) A set in R n of the form { x R n | α 1 x 1 + · · · + α n x n = b } (7) is called a hyperplane in R n . A set in R n of the form { x R n | α 1 x 1 + · · · + α n x n b } (8) is called a half-space in R n , i.e. a half-space consists of all the points on one side of a hyperplane. Thus, the set S of feasible solutions is the intersection of a finite number of half-spaces. For example, a line such as x 1 + x 2 = 1 is a hyperplane in R 2 and the inequality x 1 + x 2 1 is a half-space. The set of solutions to x 1 + x 2 1 , x 1 0 , x 2 0 is the intersection of three half-spaces. Similarly, a plane such as x 1 + x 2 + x 3 = 1 is a hyperplane in R 3 , and the set of solutions to x 1 + x 2 + x 3 1 , x 1 0 , x 2 0 , x 3 0 is the intersection of four half-spaces. There are two convenient names that we will introduce. These are multi-dimensional versions of polygons in the plane. The intersection of a finite number of half-spaces is called a convex polytope . A bounded convex polytope is called a convex polyhedron . We give three examples (you should graph all three): 1. An example of an empty feasible set x 1 + x 2 1 - x 1 ≤ - 2 - x 2 ≤ - 2 and x 1 0 , x 2 0 1
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2. An example of a convex polyhedron for the feasible set x 1 + x 2 1 and x 1 0 , x 2 0 3. An example of an unbounded convex polytope for the feasible set - x 1 + x 2 1 1 2 x 1 - x 2 1 and x 1 0 , x 2 0
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