Homework 7

Homework 7 - ( A ) < n (where n is the number of...

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CO350 LINEAR OPTIMIZATION - HW 7 Due Date: FRIDAY March 9th at the beginning of class. Please do NOT leave the assignments in the drop off box but return them to the instructor in class. From the notes: exercise 4.8.9 Exercise 1. Consider three points x 1 ,x 2 ,x 3 R n . We define Δ( x 1 ,x 2 ,x 3 ) = { λ 1 x 1 + λ 2 x 2 + λ 3 x 3 : λ 1 + λ 2 + λ 3 = 1 1 2 3 0 } . Note that, geometrically, Δ( x 1 ,x 2 ,x 3 ) corresponds to a triangle with vertices x 1 ,x 2 ,x 3 . (a) Consider a set C R n . Show that, if for all x 1 ,x 2 ,x 3 C , we have Δ( x 1 ,x 2 ,x 3 ) C , then C is convex. (b) Consider a set C R n . Show that, if C is convex, then for all x 1 ,x 2 ,x 3 C , we have Δ( x 1 ,x 2 ,x 3 ) C . Exercise 2. Consider the set B := { ( x 1 ,x 2 ) : x 2 1 + x 2 2 1 } . Note that B is the 2 -dimensional unit disk. Give an algebraic proof that B is convex. Exercise 3. Let P = { x ∈ < n : Ax b } be the set of feasible solutions to some linear program. Note that the variables in P are free. (1) Show that if rank
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Unformatted text preview: ( A ) < n (where n is the number of variables) then P has no extreme points. (2) EXTRA CREDIT (HARDER). Show that if rank ( A ) = n then P has at least one extreme point. Exercise 4. Let A be an m × n matrix and let d and b be m dimensional vectors. Consider the following statements, (S1) There is no n-dimensional non-negative vector x such that d ≤ Ax ≤ b , (S2) There are m-dimensional non-negative vectors z and y such that y T A ≥ z T A and y T b < z T d . In this question you have to, (1) Give a self contained proof that (S2) implies (S1). (2) Using the duality theorem and duality theory, give a proof that (S1) implies (S2). HINT: Use the same strategy as for the proof of Farkas’ Lemma. 1...
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This note was uploaded on 07/28/2009 for the course CO 350 taught by Professor S.furino,b.guenin during the Winter '07 term at Waterloo.

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