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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....
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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.
 Spring '10
 TanBanPin
 Linear Programming, Geometry

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