HOMEWORK 5
Due November 12
For students registered for 3 credits: 4.4, 4.7, 4.15 in the textbook, and 6.2, 6.7,
6.8, 6.9 (a-e), 6.23, 6.27 in the attached pages.
For students registered for 4 credits: 4.4, 4.7, 4.15, 4.19, 4.29 in the textbook,
and 6.2, 6

HOMEWORK 2
Due September 22
For students registered for 3 credits: 2.1, 2.9, 2.10, 2.12, (1), (2).
For students registered for 4 credits: 2.1, 2.6, 2.9, 2.10, 2.12, 2.13, (1), (2) and
(3).
(1) Find all extreme points and extreme directions of the followin

Homework 5 Solution
Exercise 4.4
Note that the primal and dual problems are of the forms
Minimize cT x
Maximize cT x
Subject to Ax c
Subject to AT x c
x 0,
x 0.
if A = AT , then x is a primal feasible solution and also dual feasible by assumption. Moreove

Duality Theory
Duality
Consider the following Linear Program Min - 4 x1 - x2 - 5 x3 - 3 x4 subject to : - x1 + x2 + x3 - 3 x4 -1 (1) -5 x1-x2 -3 x3-8 x4 -55 (2) x1-2 x2 -3 x3 + 5 x4 -3 (3) x1,x2 ,x3 ,x4 0 We wish to give a quick lower bound on the optima

HOMEWORK 4
Due October 29
(1) Solve the following problem using the revised simplex method.
min x1 + 6x2 7x3 + x4 + 5x5
1
3
s.t. x1 4 x2 + 2x3 4 x4 = 5
1
3
4 x2 + 3x3 4 x4 + x5 = 5
x1 , x2 , x3 , x4 , x5 0.
(2) Show that the following two problems are eq

HOMEWORK 3
Due October 8
For students registered for 3 credits: 3.7, 3.12, 3.17, 3.19, 3.20 and 3.31.
For students registered for 4 credits: 3.7, 3.9, 3.12, 3.17, 3.18, 3.19, 3.20, 3.23,
and 3.31.
For all students, solve problem 3.17 using AMPL (in additi

Homework 3 Solution
Exercise 3.7
(P0 ) Minimize
Subject to
cT x
(P1 ) Minimize
Ax = b
Subject to
x 0.
cT d
Ad = 0
di 0 i Z.
: Given an optimal solution of (P0 ) x , we have cT x cT x i.e.
cT (x x) 0.
(1)
Because both x and x are feasible for (P0 ), we hav

HOMEWORK 1
Due September 8
For students registered for 3 credits: 1.4, 1.8, 1.10, 1.11, 1.15, 1.17 and (1).
For students registered for 4 credits: 1.5, 1.7, 1.10, 1.11, 1.12, 1.15, 1.17, (1) and
(2).
(1) (Two-Person Zero-Sum Game) Consider the Rock-Scisso

Solution 2
Exercise 2.1
(a) This is a quarter, namely, the intersection of the orthant cfw_(x.y)|(x, y)
0 with the unit circle cfw_(x, y)|x2 +y 2 1, and is not a polyhedron. One way
of proving formally that the quarter-circle is not a polyhedron, is to n

Homework 1 Solutions
Exercise 1.4
minimize 2x1 + 3z1
s.t. z2 + z3 5
x2 10 z1
x2 + 10 z1
x1 + 2 z2
x1 2 z2
x2 z3
x2 z3
1
Exercise 1.8
Let us choose as our objective to minimize the maximum deviation denoted
by maxi |Ii Ii | of the actual illuminations f

Revised Simplex
Original Simplex worked on tableaus
Great for hand computations but terrible for implementation Revised Simplex can obtain all Tableau entries as required less computation
Connection to Gauss-Jordan Method
Matrix Manipulations
Post-mul

IE411 Linear Optimization
Complexity and Ellipsoid Method
Efficiency of Algorithms
Question: Given a problem of a certain size, how long does it take to solve it? Two kinds of Answers:
Average case: how long for a typical problem.
Mathematically diffic

IE411 Homework 7 Due: May 3 (1) Starting from (x, w, y, z) = (e, e, e, e), and using = 1/10, and r = 9/10, compute (x, w, y, z) by hand after one step of the path following method for the following problem. max s.t. 2x1 2x1 2x1 4x1 x1 x1 + + + + + , x2 x2

HOMEWORK 2
Due February 22 For students registered for 3 credits: 3.9, 3.12, 3.17, 3.19, and 3.20. For students registered for 4 credits: 3.9, 3.12, 3.17, 3.18, 3.19, 3.20, and 3.23. For all students, solve problem 3.17 using AMPL (in addition to the manu

HOMEWORK 3
Due March 12 For students registered for 3 credits: 4.4, 4.7, 4.15, 4.19 and 4.29. For students registered for 4 credits: 4.4, 4.7, 4.15, 4.19, 4.29 and 4.38.
1

HOMEWORK 4
Due April 9 For students registered for 3 credits: 5.5, 5.6 and 5.13. For students registered for 4 credits: 5.5, 5.6, 5.13, 5.10 and the following problem (a). (a) Consider n retailers each of which faces a Newsvendor problem. That is, each re

HOMEWORK 5
Due April 16 For students registered for 3 credits: (1)-(2) below. For students registered for 4 credits: (1)-(3) below. (1) Solve the following problem by the decomposition technique using two convexity constraints: max s.t. 3x1 2x1 -x1 3x1 +

IE411 Linear Optimization
Interior Point Methods
Interior Point Methods: the Breakthrough
The Wall Street Journal Waits Till 1986
AT&T Patents the Algorithm, Announces KORBX
What Makes LP Hard?
Matrix Notation
Optimality Conditions
The Interior Point Para

IE 411
Optimization of Large-Scale Linear Systems
Xin Chen
Spring 2010
Course Information
Lecture hours: MWF 2-2:50 pm Classroom: 203 TB Instructor: Professor Xin Chen Office: 216C TB Phone: 244-8685 Office hours: M 11am-12pm W 3pm-4pm Email: [email protected]

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