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
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 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 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
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 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 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
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 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
linear optimization.max
linear optimization.max
linear optimization.max
linear optimization.max
linear optimization.max
linear optimization.max
linear optimization.max
linear optimization.max
linear optimization.max
linear optimization.max
linear optimiza
MS-E2140 Linear Programming
Exercise 5
Thu 24.09.2015
Y346
Week 3
This weeks homework https:/mycourses.aalto.fi/mod/folder/view.php?id=39960 is due no
later than Tuesday 06.10.2015 23:55.
Exercise 5.1 Tableau
Course book Exercise 3.20
Consider a linear pr
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
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
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]
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
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 +
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 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 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
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 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