Math 407
Linear Optimization
The Two Phase Simplex Algorithm
Solve the following LPs using the two phase simplex algorithm.
1.
maximize
subject to
3x1
x1
x1
2x1
+
+
0
x2
x2 1
x2 3
x2
4
x1 , x 2
Solution: (1, 2)
2.
maximize
subject to
3x2
x1
2x1
3x1
0
+
Lecturer: Thomas Rothvoss
Due date: Friday, April 28, 2017, in class
Problem Set 4
409 - Discrete Optimization
Spring 2017
Exercise 1
Consider the following network with a directed graph G = (V, E), capacities u(e) (the labels of the
edges), a source s an
Lecturer: Thomas Rothvoss
Due date: Friday, April 14, 2017, in class
Problem Set 2
409 - Discrete Optimization
Spring 2017
Exercise 1
Consider the undirected graph G = (V, E) with edge cost c(e) for e E that you can see below.
i) Compute a minimum spannin
Lecturer: Thomas Rothvoss
Due date: Friday, April 21, 2017, in class
Problem Set 3
409 - Discrete Optimization
Spring 2017
Exercise 1
Consider the following directed graph G = (V, E) (edges are labelled with edge cost c(e).
3
s
2
3
1
b
3
a
c
2
2
2
d
4
e
a
Discrete Optimization
Spring 2017
Thomas Rothvoss
B
D
A
C
E
Last changes: March 24, 2017
2
Contents
1 Introduction to Discrete Optimization
1.1 Algorithms and Complexity . . . . . . . . .
1.1.1 Complexity theory and NP-hardness
1.2 Basic Graph Theory . .
Lecturer: Thomas Rothvoss
Due date: Friday, May 12, 2017, in class
Problem Set 5
409 - Discrete Optimization
Spring 2017
Exercise 1
Consider the linear program
max x1 2x2
x1 + x2
x1 3x2
2x1 x2
4x1 3x2
(P)
4
6
3
15
a) Draw the feasible region of LP (P).
b)
409 - Midterm exam - Study guide - Spring 2015
FAQ
What can and should I bring to the exam?
You should bring pens and maybe a ruler. If you like, you are allowed to bring a calculator
(though I doubt that it will be needed). You can bring a 1 page sheet
Math 409 Midterm exam
May 4, 2015
Name:
NetID:
INSTRUCTIONS READ THIS NOW
OFFICIAL
USE ONLY
This test has 5 problems worth a total of 50 points. Look over your test
booklet right now. If you find any missing pages or problems please ask
1.
me for another
Math 409 Midterm exam
May 4, 2015
INSTRUCTIONS READ THIS NOW
Name:
NetID:
OFFICIAL
USE ONLY
This test has 5 problems worth a total of 50 points. Look over your test
booklet right now. If you find any missing pages or problems please ask
1.
me for another
Math 409 Test exam
Feb 7, 2014
Name:
NetID:
INSTRUCTIONS READ THIS NOW
OFFICIAL
USE ONLY
This test has 5 problems worth a total of 100 points. Look over your test
booklet right now. If you find any missing pages or problems please ask
1.
me for another b
Math 409 Test exam
Feb 7, 2014
Name:
NetID:
INSTRUCTIONS READ THIS NOW
OFFICIAL
USE ONLY
This test has 5 problems worth a total of 100 points. Look over your test
booklet right now. If you find any missing pages or problems please ask
1.
me for another b
Lecturer: Thomas Rothvoss
Due date: Friday, May 26, 2017, in class
Problem Set 7
409 - Discrete Optimization
Spring 2017
Exercise 1
Run the Branch & Bound algorithm to solve the following integer linear program
max 3x1 + x2
2x1 + 3x2
10x1 + 4x2
x1
x2
x1 ,
Lecturer: Thomas Rothvoss
Due date: Friday, May 19, 2017, in class
Problem Set 6
409 - Discrete Optimization
Spring 2017
Exercise 1
2 . Consider the linear program:
Let G = (V, E) be a bipartite graph with parts V = V1 V
min yu
uV
yu + yv 1
yu 0
cfw_u, v
Lecturer: Thomas Rothvoss
Due date: Friday, April 7, 2017, in class
Problem Set 1
409 - Discrete Optimization
Spring 2017
Exercise 1
Let G = (V, E) be any undirected graph. Recall that deg(v) gives the degree of v V (which is the
number of edges incident
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1. Introduction to Discrete Optimization
In nite dimensional optimization we are interested in locating solutions to the problem
P : minimize
xX
subject to
f0 (x)
x .
where X is the variable space (or decision space), f0 : X R cfw_ is called the objective
Problem Set 1: Modeling Integer Programming Problems
(1) Suppose that you are interested in choosing to invest in one or more of 10 investment
opportunities. Use 0-1 variables to model the following linear constraints.
(a) You cannot invest in all opportu
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v2
e52
e12
e24
e32
v1
v4
v5
e35
e12
e13
e45
v3
Figure 1. A graph with 5 vertices.
1. Graphs, Digraphs, and Networks
1.1. The Basics. A graph is a mathematical structure comprised of two classes of objects:
vertices and edges. If we let G denote the graph,
Problem Set 2: The Branch and Bound Algorithm
(1) Solve the following integer knapsack problems by the branch and bound algorithm.
(a)
maximize 8x1 + 11x2 + 6x3
subject to 5x1 + 7x2 + 4x3 14
x cfw_0, 13
(b)
maximize 10x1 + 12x2 + 7x3
subject to 4x1 + 5x2
Problem Set 6: Flows in a Digraph
(1) Apply the ow augmenting path algorithm to solve the max-ow problems given by
the capacitated digraphs below. In each graph the source node is the node number 1
and the sink node is the highest numbered node. The arc c
v2
e12
e24
e32
v1
e13
v3
e52
v4 e45
e35
e12
v5
Figure 1. A graph with 5 vertices.
v2
a12
a32
v1
a13
v3
a24
a52
v4 a45
a35
a12
v5
Figure 2. A digraph with 5 nodes.
Problem Set 5: Graphs and Digraphs
(1) What are the degrees of all vertices in the graph in
1. Easy Interger LPs
Consider the linear integer program
IP minimize cT x
subject to Ax = b, 0 x , and x Zn ,
where c Rm , A Zmn , and b Zm . The LP relaxation of IP is the linear program
IP relax minimize cT x
subject to Ax = b, 0 x .
We say that the a s
e3=(2,4)
2
4
e6=(4,6)
e1=(1,2)
e5=(3,2)
e8=(5,4)
e4=(2,5)
1
6
e2=(1,3)
e1=(1,2)
e9=(5,6)
e7=(5,3)
5
3
Figure 1. A simple network
1. Network Flows
1.1. Flows. Let G = (V, E) be a network with |V | = n and |E| = m. A ow f on G is any
function of the arcs E