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 alte
FINAL EXAM SAMPLE PROBLEM PARTIAL SOLUTIONS FOR MATH 407
3. Solve the following LP stating its solution and optimal value.
maximize 4x1
subject to x1
x1
2x2
3x1
0
+ 4x2 +
+ x2 +
+ x2 +
+ 2x2 +
+ 2x2 +
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
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 t
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 lin
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
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
c
Math 407 Section A
SAMPLE PROBLEMS FOR THE FIRST QUIZ
1. Consider the system
4x1
x3 = 200
9x1 + x2 x3 = 200
7x1 x2 + 2x3 = 200 .
Solution
x1
30
x2 = 150
x3
80
2. Represent the linear span of the fo
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 sens
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
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 <
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