Chapter 2 - MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 2:...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 2: Geometry of Linear Programming 1 / 30 A polyhedron or polyhedral set can be described in the form { x ∈ R n | Ax ≥ b } , where A is an m × n matrix and b is a vector in R m . Geometrically, a polyhedron is a finite intersection of half spaces a i x ≥ b i . A bounded polyhedron is a polytope . The feasible region for a standard form LP { x ∈ R n | Ax = b , x ≥ } is also a polyhedron. 2 / 30 In this topic, we characterize the corner points of polyhedron geometrically (via extreme points and vertices) and algebraically (via basic feasible solutions). The main result states that a nonempty polyhedron has at least one corner point if and only if it does not contain a line, and if this is the case, the search for optimal solutions to linear programming problems can be restricted to corner points. 3 / 30 Extreme Point, Vertex and Basic Feasible Solution • x is a convex combination of x 1 ,..., x k iff x = ∑ k i =1 λ i x i , where the scalars λ i ∈ [0 , 1] are such that ∑ k i =1 λ i = 1. • The convex hull of x 1 ,..., x k is the set CH ( x 1 ,..., x k ) = ( x ∈ R n x = k X i =1 λ i x i , λ i ∈ [0 , 1] , k X i =1 λ i = 1 ) . 4 / 30 Terminology 1 If a vector x * ∈ R n satisfies the constraint a i x T b i at equality; that is a i x * = b i , then the corresponding constraint a i x T b i is said to be active (or binding ) at x * . 5 / 30 Three definitions of the concept of corner point Consider a polyhedron P ⊂ R n defined by a set of linear equality and inequality constraints. (a) A vector x * ∈ P is an extreme point of P if we cannot find two vectors y , z ∈ P both different from x , and a scalar λ ∈ [0 , 1], such that x * = λ y + (1- λ ) z . (Geometric definition) (b) A vector x * ∈ P is a vertex of P if we can find some c ∈ R n such that c x * < c y for all y ∈ P and y 6 = x * . (Geometric definition) (c) A vector x * ∈ P is a basic feasible solution if there exist n linearly independent constraints that are active at x * . (Algebraic definition) 6 / 30 Terminology 1 A vector x * ∈ R n is said to be of rank k , if the set { a i | a i x * = b i } contains k , but not more than k , linearly independent vectors. In other words, the span of { a i | a i x * = b i } has dimension k . Thus, a vector x * ∈ P is a basic feasible solution if and only if it has rank n . 2 A vector x * ∈ R n (not necessary in P ) is a basic solution if there are n linearly independent vectors in the set { a i | a i x * = b i } . Moreover, every equality constraint (if any) must be satisfied at a basic solution....
View Full Document

This note was uploaded on 12/02/2011 for the course MATH 3252 taught by Professor Tanbanpin during the Spring '10 term at National University of Singapore.

Page1 / 30

Chapter 2 - MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 2:...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online