Math 407 Linear Optimization
1
Introduction
1.1
What is optimization?
A mathematical optimization problem is one in which some function is either maximized or
minimized relative to a given set of alternatives. The function to be minimized or maximized
is
FINAL EXAM OUTLINE FOR MATH 407
EXAM DATES:
Section A: Monday, December 12, 2015: 8:30 -10:20am.
Section B: Wednesday, December 14, 2015: 2:30 -4:20pm.
EXAM OUTLINE
The final exam will consist of 6 questions each worth 60 points (except for problem 2 wh
Solutions to Homework 3
The following is Question 4 from
http:/www.math.washington.edu/burke/crs/407/suppl/simplex2.pdf
Solve the following LP using the two-phase simplex algorithm
maximise
subject to
x1
+ x2 + x3
2 3
1
x1
1
3 1 x 2
2 1 3
x3
Solution:
1
Duality Theory
Recall from Section 1 that the dual to an LP in standard form
cT x
Ax b, 0 x
maximize
subject to
(P )
is the LP
bT y
AT y c, 0 y.
minimize
subject to
(D )
Since the problem D is a linear program, it too has a dual. The duality terminology
1
Solutions to homework 5
1. Let A Rmn and
M
A
I
=
R(mn)n
Let vi denote the i-th row of the matrix M and S cfw_1, . . . , m + n. Show
that c Cone(vi | i S ) if and only if c = AT y r for some y
0
and r
0 with yi = 0 if i S cfw_1, . . . , m and rj = 0 if
Solutions to Homework 4
The following is Question 4 from
http:/www.math.washington.edu/burke/crs/407/suppl/cs.pdf
T
Is x = (0, 0, 0, 0, 0, 10) optimal for the following LP P :
maximise
subject to
T
cT x
Ax
b, x
0
T
where c = (2, 4, 1, 0, 6, 8) , b = (10,
1
Sensitivity Analysis
In this section we study general questions involving the sensitivity of the solution to an LP
under changes to its input data. As it turns out LP solutions can be extremely sensitive to
such changes and this has very important pract
Math 407 Section A
MATH 308 REVIEW
In this course the notion of linearity plays a central role. All of the theoretical aspects of
this course are based on properties of systems of linear equations and inequalities in I n . For
R
this reason the course pre
MATH 407
LA Review
Computing Solutions sets to Ax = b when m < n
A number of students have asked me to describe the procedure for representing the set of solutions to the
linear system Ax = b when m < n. I will give a example illustrating this procedure b
1
LP Geometry
We now briey turn to a discussion of LP geometry extending the geometric ideas developed
in Section 1 for 2 dimensional LPs to n dimensions. In this regard, the key geometric idea
is the notion of a hyperplane.
Denition 1.1 A hyperplane in R