{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
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 } bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright 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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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 finite 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
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Proof of Extreme Point Optimality 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 summationdisplay i =1 λ i c T x i > I summationdisplay i =1 λ i c T x * = c T x * and we have a contradiction. 4
Image of page 4
Unbounded Polyhedra A feasible direction of an unbounded polyhedra X R n is a (non-zero) vector d R n , such that if x 0 X then ( x 0 + λd ) X for all λ 0.
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}