(4) Recall that in lecture we have seen the theorem (without proof) that the BFS are the same as the extreme points. In this question you will prove that (under mild assumptions on P) any BFS is an extreme point. Let P = {x = R¹ | Ax b, x ≥ 0} be a polyhedron in canonical form and assume P has no degenerate BFS. Let x* € P be a BFS. Let N = {i | x is a non-basic variable} be the set of n − m non-basic variables at x*. Note that x = 0 = all i E N and that x‡ ‡ 0 for all i ‡ N since x* is non-degenerate. Suppose by contradiction that x* is not an extreme point so x* = λy + (1 – X)z for some distinct points y, z € P and λ € (0, 1). (a) Show that yį = Zį = 0 all i ɛ N. (b) Show that y and z are BFS. Hint: which constraints are active at these points? (c) Show that y = z = x. Conclude that x* is an extreme point.
(4) Recall that in lecture we have seen the theorem (without proof) that the BFS are the same as the extreme points. In this question you will prove that (under mild assumptions on P) any BFS is an extreme point. Let P = {x = R¹ | Ax b, x ≥ 0} be a polyhedron in canonical form and assume P has no degenerate BFS. Let x* € P be a BFS. Let N = {i | x is a non-basic variable} be the set of n − m non-basic variables at x*. Note that x = 0 = all i E N and that x‡ ‡ 0 for all i ‡ N since x* is non-degenerate. Suppose by contradiction that x* is not an extreme point so x* = λy + (1 – X)z for some distinct points y, z € P and λ € (0, 1). (a) Show that yį = Zį = 0 all i ɛ N. (b) Show that y and z are BFS. Hint: which constraints are active at these points? (c) Show that y = z = x. Conclude that x* is an extreme point.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.4: Mathematical Induction
Problem 46E
Related questions
Question
Transcribed Image Text:(4) Recall that in lecture we have seen the theorem (without proof) that the BFS are the same
as the extreme points. In this question you will prove that (under mild assumptions on P)
any BFS is an extreme point. Let P = {x = R¹ | Ax b, x ≥ 0} be a polyhedron in
canonical form and assume P has no degenerate BFS. Let x* € P be a BFS. Let N = {i |
x is a non-basic variable} be the set of n − m non-basic variables at x*. Note that x = 0
=
Transcribed Image Text:all i E N and that x‡ ‡ 0 for all i ‡ N since x* is non-degenerate. Suppose by contradiction
that x* is not an extreme point so x* = λy + (1 – X)z for some distinct points y, z € P and
λ € (0, 1).
(a) Show that yį = Zį = 0 all i ɛ N.
(b) Show that y and z are BFS. Hint: which constraints are active at these points?
(c) Show that y = z = x.
Conclude that x* is an extreme point.
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