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
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
+
i fgf p x x x y p fe 1PtQQqswte j tol D8lz1o1su xheswh2d v v v v v r 2z w tv tz w v 2x v v w v 2 w tz tu w 8u 2z v v v v w v x v w tv w tz 2z 8u v v w v u 2z w tz t w 2x v sedpDsi| t v v v v w p f v 2x w tz t w 2 v hshw Qq v v v w i w tu r i fgf p 2r h
MATH 409 SAMPLE QUIZ
NAME (Please print):
April 1, 2011
There are 2 problems. Stop now and make sure you have both problems. If you do not have them
both, then request a new quiz. Both problems are worth 35 points for a total of 70 points. Show all
of you
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 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
Problem Set 7: The Network Simplex Algorithm
Draw the network and solve the network transhipment problem
min cT x
subject to Ax = b, 0 x
with
1 1
1
1
1
1
1 1 1
1
1 1
A=
1
1 1
1
1
1
1
1
1
1
1
1
using the network simplex algorithm and the following 3 choic
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
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,
lcn yb fff b b b m m TpxTy2gfqTejpgte EgrfhB n rf xv f rd r q rt W t vj t r q v g Ev frGg1Tp y ph Tr x f i r y g g v r kreshpppst2pep#f12hi t [email protected] pBTEx f f4fqx krpgtr fhTiqgq jfkr#eptv jB f i h q f c q h g r f t 2 hgtq ht t x r rui u qru u v r qv
e l o nl o n l o 2n a l o q o n l o | o l o l o n o o
e l o
$ " ! 43210)('&%#
y Eer rq rq tkyasr f s r s r gVRfuVs2Q w z w vh X TR reatkr xr o q f l n l d f n f 2n 2pl l l X j h d 2e `RgkXeWi agsr f #caVh Wf ff a T rx b f fTx WY v h X TR SVWuVhtfe4W1c
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