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 m
494
CHAPTER 9
LINEAR PROGRAMMING
9.3 THE SIMPLEX METHOD: MAXIMIZATION
For linear programming problems involving two variables, the graphical solution method introduced in Section 9.2 is convenient. Ho
Math 407A: Linear Optimization
Professor James Burke
Math Dept, University of Washington
Linear Algebra Review
Professor James Burke (Math Dept, University of Washington) Linear Optimization
Math 407A
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
Math 407
Linear Least Squares Homework
This homework set will focus on the linear least squares problem
LLS
minn
xR
1
kAx bk2 ,
2
where A Rmn and b Rm .
(1) Listed below are two functions. In each cas
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
Linear Programming
Lecture 2: Introduction to Linear Programming
1
Author: James Burke, University of Washington
1
Math 407: Introduction
What is linear programming?
Applications of Linear Programing
Linear Programming
Lecture 1: Linear Algebra Review
Lecture 1: Linear Algebra Review
Linear Programming
1 / 24
1
Linear Algebra Review
2
Linear Algebra Review
3
Block Structured Matrices
4
Gaussian El
Introduction to Game Theory
Matrix Games and Lagrangian Duality
1. Introduction
In this section we study only finite, two person, zero-sum, matrix games. We introduce the basics
by studying a Canadian
4
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
minimize
subject to
(D)
c, 0 y.
Since the problem D is a linea
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 circu
2
2.1
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 si
MATH 407 Key Theorems
Theorem 0.1 (Weak Duality Theorem). If x Rn is feasible for P and y Rm is feasible for D, then
cT x y T Ax bT y.
Thus, if P is unbounded, then D is necessarily infeasible, and if
MATH 407 FINAL EXAM SAMPLE QUESTIONS
1. A farmer has three farms. He can grow three dierent crops on them. Data for the coming season
are given in the table below. To maintain a uniform work load amon
SAMPLE QUESTIONS FOR FINAL
1. Model the following two problems as LPs.
(a) A company needs to lease warehouse storage space over the next 5 months. Just how
much space will be required in each of thes
FINAL EXAM OUTLINE FOR MATH 407
EXAM DATE: Monday, December 8, 2014: 8:30am-10:20am
The nal exam for this course is set to be given on Monday, December 8, 2014: 8:30am-10:20am
in the same classroom th
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 th
THE LINEAR LEAST SQUARES PROBLEM
1. Introduction
A linear least squares problem is one of the form
2
1
minimize
2 kAx bk2 ,
n
(1)
xR
where
A Rmn , b Rm ,
2
2
.
kyk2 := y12 + y22 + + ym
and
Problems of
Math 407A: Linear Optimization
Lecture 8: Initialization and the Two Phase Simplex Algorithm
Math Dept, University of Washington
Initialization
We have shown that if we are given a feasible dictionary
FINAL EXAM OUTLINE FOR MATH 407
EXAM DATE: Thursday, August 18, 2011: 9:40am-11:40am
The nal exam for this course is set to be given on Thursday, August 18 at 9:40am-11:40am in
the same classroom that
486
CHAPTER 9
LINEAR PROGRAMMING
9.2 LINEAR PROGRAMMING INVOLVING TWO VARIABLES
Many applications in business and economics involve a process called optimization, in
which we are required to find the
SECTION 9.4
32. The accounting firm in Exercise 31 raises its charge for an
audit to $2500. What number of audits and tax returns will
bring in a maximum revenue?
35. (Maximize)
Objective function:
z
SECTION 9.5
9.5
THE SIMPLEX METHOD: MIXED CONSTRAINTS
521
THE SIMPLEX METHOD: MIXED CONSTRAINTS
In Sections 9.3 and 9.4, we looked at linear programming problems that occurred in standard form. The co
Math 407A: Linear Optimization
Lecture 3: LP Modeling
Math Dept, University of Washington
LP Modeling
Model 10: Detergent Production
Model 9: Investing Over Time
Model 4: Blending
LP Modeling
The four
Math 407A: Linear Optimization
Lecture 11: The Dual Simplex Algorithm
Math Dept, University of Washington
The Dual Simplex Algorithm
P
maximize
subject to
4x1 2x2 x3
x1 x2 + 2x3 3
4x1 2x2 + x3 4
x1 +
Math 407A: Linear Optimization
Lecture 5: Simplex Algorithm I
Math Dept, University of Washington
1
Dictionaries, Augmented Matrices, the Simplex Tableau
Dictionaries
The Simplex Tableau
2
Basic Feasi
Linear Programming
Lecture 7: Does the Simplex Algorithm Work?
Math Dept, University of Washington
Does the Simples Algorithm Work?
Choosing Entering and Leaving Variables
Unbounded LPs
Degeneracy
Ove
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