Question

Linear Programming: Extreme points and extreme directions of a convex set!

Asked by: Euroman_21

Given a set D  (a,b,c,d,e,f)^T in  R^6

which statisfies that:

3a + b - c + 2d + 3e - f = 2

a  - b + 2c - 4d + 8e + f = 5

Find all extreme points and extreme directions for D.

I can see that if add the equations together I get:

)) 4a - c - 2d + 11e = 7,

Thusly b and f free variables. Aren't they they? Which would allow me to choose either of the two

Then I solve the )) for e this gives e = [7-4a-c+2d]/11.

By Inserting e into the second orignal equation, I get
f= 5- a+ b- 2c- 4d- (8/11)(7- 2a-2c+ 2b)= -1/11- (5/11)a- b+ (6/11)c- (28/11)d.

But from this how do I get the set of all possible extreme points for D???

Secondly the extreme directions of the set D. How do I find them??

I know that two direction of a closed convex set can be expressed as:

d = lamda_1 * d_1 +  lamda_2 * d_2

where lambda_1, lambda_2 > 0

Could it that if we say that let d be the span of D, then the set of all extreme direction is an unique vector lamda_a which saties x + \lamda_a * d ?? where x being a abitrary extreme point of D??

Hope Somebody can give a hint ?

Sincerely Yours
Fred

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Asked On
2007-06-24 at 08:46:38ID22654183
Tags

linear

,

programming

,

extreme

Topic

Math & Science

Participating Experts
1
Points
50
Comments
1

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Answers

 

by: sgvillPosted on 2007-07-05 at 10:32:34ID: 19425747

As a possible hint, notice that if you multiply the first equation by 2 and add it to the second, you have a different set of free variables...

20120131-EE-VQP-002

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