# hw1 - (e) An unbounded n-dimensional polyhedral set can...

This preview shows page 1. Sign up to view the full content.

HOMEWORK 1 Due February 8 For students registered for 3 credits: 1.3, 1.10, 1.11, 1.15, 2.1, 2.9, (1), (2), and (3). For students registered for 4 credits: 1.5, 1.7, 1.10, 1.11, 1.12, 1.15, 2.1, 2.6, 2.9, (1), (2) and (3). (1) Find all extreme points and extreme directions of the following polyhedral set: X = { ( x 1 ,x 2 ,x 3 ,x 4 ) : x 1 ,x 2 ,x 3 ,x 4 0 , - x 1 + x 2 - 2 x 3 1 , - 2 x 1 - x 3 + 2 x 4 2 } (2) Answer the following questions and provide a brief explanation or illustra- tion. (a) Is it possible for X = { x : Ax b,x 0 } to be empty but D = { d : Ad 0 , 1 d = 1 ,d 0 } to be nonempty? (b) Is there a relationship between redundancy and degeneracy of a poly- hedral set? (c) Does degeneracy imply redundancy in two dimensions? (d) If the intersection of a ﬁnite number of halfspaces is nonempty, then this set has at least one extreme point. True or false?
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (e) An unbounded n-dimensional polyhedral set can have at most n ex-treme directions. True or false? (f) What is the maximum (actual) dimension of X = { x : Ax = b,x ≥ } , where A is m × n of rank t , t ≤ m ≤ n ? (3) Consider the following table production problem. Table one can be sold at 10 per unit, which requires 2 units of wood one and 1 unit of wood two. Similarly, Table one can be sold at 14 per unit, which requires 1 units of wood one and 2 unit of wood two. There are three agents each of which controls a resource vector (2 , 4), (4 , 2) and (4 , 4) respectively. Describe the core of the resulting cooperative game. 1...
View Full Document

## This note was uploaded on 09/08/2010 for the course IESE IE411 taught by Professor Xinchen during the Fall '09 term at University of Illinois, Urbana Champaign.

Ask a homework question - tutors are online