3_lp_geometry - Geometry of LPs Consider the following LP:...

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Geometry of LPs Consider the following LP * : min { c T x | a T i x b i i = 1 ,...,m } . The feasible region is X := { x R n | a T i x b i i = 1 ,...,m } = m i =1 { x R n | a T i x b i } b ² X i The set X i is a Half-space . The set H i = { x R n | a T i x = b i } is a Hyperplane . The feasible region X is given by the intersection of m half-spaces and is known as a Polyhedron . A Polyhedron is a closed convex set (verify?). If it is bounded it is called a Polytope . LP: Minimize a linear function over a polyhedral set. Move in the direction - c as far as possible while staying within the feasible region X . * Verify that any LP can be written in this form 1
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Example min c 1 x 1 + c 2 x 2 s . t . - x 1 + x 2 1 , x 1 0 , x 2 0 . X1 X2 -X1 + X2 = 1 X1 X2 -X1 + X2 = 1 c = (1 , 1) T , x * = (0 , 0) T c = (1 , 0) T , x * = (0 ,x 2 ) T for 0 x 2 1 X1 X2 -X1 + X2 = 1 X1 X2 -X1 + X2 = 1 c = (0 , 1) T , x * = ( x 1 , 0) T c = ( - 1 , - 1) T for 0 x 1 Unbounded. 2
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Extreme Point Optimality An optimal solution (if it exists) always lies on a boundary of the feasible region X If there is an optimal solution, then there is one at a corner or a vertex or an extreme point of the polyhedron X . A point x X is an extreme point of X if it cannot be expressed as a convex combination of some other points x 1 ,...,x m X . A polyhedron has a Fnite number of extreme points. Representation theorem: When X is a polytope , any point x X can be expressed as a convex combination of the extreme points of X , i.e. x = I i =1 λ i x i where x i vert( X ) for all i = 1 ,...,I , I i =1 λ i = 1, and λ i 0. 3
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Consider the LP min { c T x | x X } , where X is a polytope. Let x * be an optimal solution, and suppose that there are no extreme points of X that are optimal. Then c T x * < c T x i for all x i vert( X ). By the representation thm: x * = I i =1 λ i x i , then c T x * = I s i =1 λ i c T x i > I s i =1 λ i c T x * = c T x * and we have a contradiction. 4
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This note was uploaded on 02/04/2012 for the course ECON 6673 taught by Professor Ahmed during the Spring '11 term at Georgia Institute of Technology.

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3_lp_geometry - Geometry of LPs Consider the following LP:...

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