CO 250 Assignment 6 Spring 2012
Due: Friday June 22nd by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
a
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),
CO 250 Assignment 6 Spring 2012
Grading Scheme and Common Errors
Problem 1: Simplex Geometry
Consider the LP maxcfw_cT x : Ax b, x 0, where
A :=
21
24
b :=
4
8
c :=
2
.
3
a) Convert this LP into SEF in the standard way, and then solve it via Simplex. In e
CO 250 Assignment 6 Spring 2012
Solutions
Problem 1: Simplex Geometry
Consider the LP maxcfw_cT x : Ax b, x 0, where
A :=
21
24
b :=
4
8
c :=
2
.
3
a) Convert this LP into SEF in the standard way, and then solve it via Simplex. In each iteration, for
the
CO 250 Assignment 7 Spring 2012
Due: Friday June 29th by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
a
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),
CO 250 Assignment 7 Spring 2012
Grading Scheme and Common Errors
Problem 1: Geometry of the simplex
Consider the following linear program
maximize c1 x1 + c2 x2
subject to
x1 +
x2
x1
x2
x1
x2
4
3
3
0
0
(a) Give a diagram showing the set of feasible soluti
CO 250 Assignment 7 Spring 2012
Solutions
Problem 1: Geometry of the simplex
Consider the following linear program
maximize c1 x1 + c2 x2
subject to
x1 +
x2
x1
x2
x1
x2
4
3
3
0
0
(a) Give a diagram showing the set of feasible solutions of the LP
(b) For e
CO 250 Assignment 8 Spring 2012
Due: Friday July 6th by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
a
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),
9
CO 250 Assignment 8 Spring 2012
Grading Scheme and Common Errors
Problem 1:
For each of the following points, determine whether it is an extreme point of the polytope P dened by
1
1 0 1
4
1 1 1
1 2 1 x 5
5
1 2 1
5
210
(a) (2, 1, 1)T
(b) (3, 1, 2)T
(c) (0,
CO 250 Assignment 8 Spring 2012
Solutions
Problem 1:
For each of the following points, determine whether it is an extreme point of the polytope P dened by
1 0 1
1
4
1 1 1
1 2 1 x 5
5
1 2 1
210
5
(a) (2, 1, 1)T
(b) (3, 1, 2)T
(c) (0, 1, 3)T
(d) (1, 3, 0)T
CO 250 Assignment 9 Spring 2012
Due: Friday July 13th by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
a
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),
CO 250 Assignment 9 Spring 2012
Grading Scheme and Common Errors
Problem 1:
For each of (a) and (b) do the following:
Write the dual (D) of the given linear program (P).
Use Weak Duality to prove that x is optimal for (P) and y optimal for (D).
a)
max
(
CO 250 Assignment 9 Spring 2012
Solutions
Problem 1:
For each of (a) and (b) do the following:
Write the dual (D) of the given linear program (P).
Use Weak Duality to prove that x is optimal for (P) and y optimal for (D).
a)
max
(2, 1, 0)x
s.t.
1 32
x
1
CO 250 Assignment 10 Spring 2012
Due: Friday July 20th by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
a
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),
CO 250 Assignment 5 Spring 2012
Solutions
Problem 1: Simplex I Consider the following linear programming problem. Start with the basis B := cfw_1, 2, put the problem in canonical form and solve the problem using the Simplex Method. max z = 5x1 + x2 - x3 -
CO 250 Assignment 5 Spring 2012
Grading Scheme and Common Errors
Problem 1: Simplex I
Consider the following linear programming problem. Start with the basis B := cfw_1, 2, put the problem
in canonical form and solve the problem using the Simplex Method.
CO 250 Assignment 1 Spring 2012
Due: Friday May 11th by 10:30 am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.) : TBA
a
Section 2 (J. Knemann, MWF 10:30-11:50 a.m.) : TBA
o
Assignment policy: Wh
CO 250 Assignment 1 Spring 2012
Grading Scheme and Common Errors
Problem 1: Linear Algebra Review
For each of the following statements, either prove that the statement is true, or provide a counterexample
that shows that the statement is false.
(a) For tw
CO 250 Assignment 1 Spring 2012
Solutions
Problem 1: Linear Algebra Review
For each of the following statements, either prove that the statement is true, or provide a counterexample
that shows that the statement is false.
(a) For two 2 2 matrices A and B
CO 250 Assignment 2 Fall 2011
Due: Friday May 18th by 10am Solutions are due in the drop boxes outside MC 4066 by the due time. Section 1 (L. Sanit`, TTh 1-2:20 p.m.) a Box 9 Slots 6 (AD), 7 (EI), 8 (JM), 9 (NQ), 10 (RU), 11 (V Z) Section 2 (J. Knemann, M
CO 250 Assignment 2 Fall 2011
Grading Scheme and Common Errors
Problem 1: Shortest Paths I Write the integer program for the shortest s, t-path problem for the graph shown below. Do not use compact notation, but rather write out all constraints.
a s b t
c
CO 250 Assignment 2 Fall 2011
Solutions
Problem 1: Shortest Paths I
Write the integer program for the shortest s, t-path problem for the graph shown below. Do not use
compact notation, but rather write out all constraints.
a
s
b
c
t
Solution: From the cou
CO 250 Assignment 3 Spring 2012
Due: Friday May 25th by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
a
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),
9
CO 250 Assignment 3 Spring 2012
Grading Scheme and Common Errors
Problem 1: LP Solutions
Let A be an m n matrix. Prove that for each b Rm , either the system of equations Ax = b has no
solution or there exists a solution x with at most m nonzero component
CO 250 Assignment 3 Spring 2012
Solutions
Problem 1: LP Solutions
Let A be an m n matrix. Prove that for each b Rm , either the system of equations Ax = b has no
solution or there exists a solution x with at most m nonzero components.
Solution: If the sys
CO 250 Assignment 4 Spring 2012
Due: Friday June 1st by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
a
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),
9
CO 250 Assignment 4 Spring 2012
Grading Scheme and Common Errors
Problem 1: SEF
Let A, B, D be matrices and b, c, d, f be vectors (all of suitable dimensions). Convert the following LP
with variables x and y (where x, y are vectors) into SEF,
min cT x + d
CO 250 Assignment 4 Spring 2012
Solutions
Problem 1: SEF
Let A, B, D be matrices and b, c, d, f be vectors (all of suitable dimensions). Convert the following LP
with variables x and y (where x, y are vectors) into SEF,
min cT x + dT y
s.t.
Ax By b
Dy
f
y
CO 250 Assignment 5 Spring 2012
Due: Friday June 8th by 10am Solutions are due in the drop boxes outside MC 4066 by the due time. Section 1 (L. Sanit`, TTh 1-2:20 p.m.) a Box 9 Slots 6 (AD), 7 (EI), 8 (JM), 9 (NQ), 10 (RU), 11 (V Z) Section 2 (J. Knemann,
CO 250 Assignment 10 Spring 2012
Solutions
Problem 1:
(a) Write the dual (D) of the given linear program (P), and the complementary slackness conditions for
the pair (P) and (D).
max (2, 4, 2)x
s.t.
12
7
0 1 1 =
x
9 0
0
2 2 2
=
3
9
5
0
x1 , x2 0, x3 free