# Combinatorial optimization: networks and matroids by Lawler E.L.

By Lawler E.L.

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Extra resources for Combinatorial optimization: networks and matroids

Sample text

2) to be satisfied, since the nonbasic slack variables s2, s3 must take on zero values. 15. 15. s, = I A SI =: I. s* = 2. s, = 2 x, =: 3. x2 = :, s2 = \$ C D B E The same situation exists in higher dimensions. That is, each basic feasible solution corresponds to an extreme point of the convex polytope of the linear program. It may, however, be the case that several basic feasible solutions correspond to the same extreme point. 1) the constraint 2x, + x2 I 5. 7) Duality Theory 53 This is simply because there are now three nonparallel straight lines intersecting at the point A, and any two of them are sufficient to determine A.

Algebraically, this condition is stated as follows. A set C is convex if x1 EC, x2 E C, 0 I /1 I 1 implies )Lxl + (1 - 2)x2 EC. A vector /Ix’ + (1 - 2)x2, where 0 I i. I 1, is said to be a convex combination of the vectors x1 and x2. It is easy to see that for any linear programming problem, any convex combination of two feasible solutions is also a feasible solution. Accordingly, the polytope defined by its inequality constraints is convex. An extreme point of a convex set is a point that is not the convex combination of any two distinct points in the set.

Accordingly, the polytope defined by its inequality constraints is convex. An extreme point of a convex set is a point that is not the convex combination of any two distinct points in the set. The extreme points of a convex polytope occur at its vertices. We shall use the terms vertex and extreme point synonomously. 17 nonconvex Example region of 52 Mathematical Preliminaries solutions of a linear programming problem and the extreme points of its convex polytope. 3) L 0. 2) to be satisfied, since the nonbasic slack variables s2, s3 must take on zero values.