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

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MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 2: Geometry of Linear Programming 1 / 30
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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 0 i x b i . A bounded polyhedron is a polytope . The feasible region for a standard form LP { x R n | Ax = b , x 0 } is also a polyhedron. 2 / 30
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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
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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
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Terminology 1 If a vector x * R n satisfies the constraint a 0 i x T b i at equality; that is a 0 i x * = b i , then the corresponding constraint a 0 i x T b i is said to be active (or binding ) at x * . 5 / 30
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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 0 x * < c 0 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
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Terminology 1 A vector x * R n is said to be of rank k , if the set { a i | a 0 i x * = b i } contains k , but not more than k , linearly independent vectors. In other words, the span of { a i | a 0 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 0 i x * = b i } . Moreover, every equality constraint (if any) must be satisfied at a basic solution. 3 A basic solution that satisfies all constraints is a basic feasible solution.
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