(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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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