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
+
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
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
3
Does the Simplex Algorithm Work?
In this section we carefully examine the simplex algorithm introduced in the previous chapter.
Our goal is to either prove that it works, or to determine those circumstances under which it
may fail. If the simplex does n
1
1.1
Solving LPs: The Simplex Algorithm of George Dantzig
Simplex Pivoting: Dictionary Format
We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example. Consider the LP (1.1) max 5x1 + 4x2 + 3x3
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
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
Math 407A: Linear Optimization
Lecture 10: General Duality Theory
Math Dept, University of Washington
ecture 10: General Duality Theory (Math Dept, University of Washington)
Math 407A: Linear Optimization
1 / 17
1
General Duality Theory
2
General Weak Dua
MATRICES, BLOCK STRUCTURES AND GAUSSIAN ELIMINATION
Numerical linear algebra lies at the heart of modern scientic computing and computational science. Today it
is not uncommon to perform numerical computations with matrices having millions of components.
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
Linear Programming
Lecture 13: Sensitivity Analysis
Sensitivity Analysis
Silicon Chip Corporation
Break-even Prices and Reduced Costs
Range Analysis for Objective Coecients
Resource Variations, Marginal Values, and Range Analysis
Right Hand Side Perturbat
Math 407A: Linear Optimization
Lecture 12: The Geometry of Linear Programming
Math Dept, University of Washington
The Geometry of Linear Programming
Hyperplanes
Denition: A hyperplane in Rn is any set of the form
H (a, ) = cfw_x : aT x =
where a Rn \ cfw
Math 407
Linear Optimization
Simplex Algorithm for Problems in Standard Form and having Feasible Origin
Solve the following LPs using the simplex algorithm in simplex tableau form. At each stage
of the simplex algorithm identify the BFS identied by the cu
Math 407
Linear Optimization
Transformation of LPs to Standard Form
Transform the following LPs to LPs in standard form.
1.
x1 12x2
minimize
subject to
5x1
2x1 +
x2 0
2.
maximize
minimize
2x3
x2
2x3 =
10
x2 20x3 30
, 1
x3 4
3x 12y +
4z
10z =
10
y 17z 1
Math 407
Denitions : Sections 13
Section 1
Mathematical Optimization: A mathematical optimization problem is one in which some real-valued
function is either maximized or minimized relative to a given set of feasible alternatives. In this
course we only
Math 407
Linear Optimization
Graphical solutions of two dimensional LPs
Solution Procedure
Step 1: Graph each of the linear constraints indication on which side of the constraint the
feasible region must lie. Dont forget the implicit constraints!
Step 2:
Math 407
Linear Optimization
Checking Optimality Via Geometric Duality
We consider LPs in standard form:
P:
maximize cT x
subject to Ax b,
0x
In the problems below, check optimality by trying to write the objective normal as a nonnegative linear combinati
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
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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 %lv2gf@ypE 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
Linear Optimization
Matrix Games
1 / 13
A Canadian Drinking Game: Morra
Each player chooses either the loonie or the toonie and places the single
coin in their closed right hand with the choice hidden from their
opponent. Each player then guesses the play
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 .
(a) Write the augmented matrix corresponding to this system.
(b) Reduce the augmented system in part (a) to echelon form.
(c)
Homework 5
Due 15th of August 2013
1. Let A Rmn and
M
=
A
In
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 r
Concrete Products Corporation
Concrete Products Corporation has the capability of producing four types
of concrete blocks. Each block must be subjected to four processes: batch
mixing, mold vibrating, inspection, and yard drying. The plant manager
desires
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
Due 1st of August
1. Review the example on the two phase simplex algorithm in Section 3.2 of
http:/www.math.washington.edu/burke/crs/407/notes/section3.pdf
This is the same example I partially wrote down in class.
2. Do questions 4 and 5 from
http:/www.ma
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,