Math 171A Homework 2 Solutions
Instructor: Jiawang Nie February 1, 2012
1. (8 points) Consider the feasible set F defined by the following constraints x1 + x2 4, x1 + 3x2 6, 6x1 - x2 18, 3 x2 6, x1 -1.
(a) Express F in the standard form Ax b. Write down A
Math 171A Homework 3 Solutions
Instructor: Jiawang Nie February 1, 2012
1. (4 points) For a given nonzero matrix A and nonzero vector b, assume that b may be written as b = bR + bN , where bR Range(A) and bN Null(AT ). (a) Show that bR and bN are unique.
Math 171A: Mathematical Programming
Instructor: Philip E. Gill
Winter Quarter 2009
Homework Assignment #2 Due Friday January 23, 2009 The starred exercises require the use of Matlab. Remember that it is necessary to do all the Matlab assignments to obtain
Math 171A: Linear Programming
Instructor: Philip E. Gill
Winter Quarter 2015
Homework Assignment #2
Due Friday January 23rd, 2015
The starred exercises require the use of Matlab. Remember that it is necessary to do all
the Matlab assignments to obtain cre
Math 140A, Fall 2010, Midterm, 11/8/10
Instructions. Answer all questions. You may use without proof anything which was proved in class. Cite a theorem either by name, if it has one, or by briefly stating what it says. 1. (20 points) Give an example of an
Math 171A Practice Midterm II
Instructor: Jiawang Nie March 5, 2012
1. Consider the LP of minimizing c 2 3 A = 4 2 3
x subject to Ax b where -15 3 5 -13 4 3 , b = -20 , c = A 5 1 -12 4 4 -13 5 2
0 1 0 . 1 1
Find the minimizer of this LP by using optimali
Math 171A: Linear Programming
Instructor: Philip E. Gill
Winter Quarter 2015
Homework Assignment #1
Due Friday January 16, 2015
I know that you are all aware of the importance of doing the homework assignments. This
is the best way to keep up with the cla
Math 171A Homework Assignment # 1
Due Date: Thursday, January 12, 2016
1. (10 points) Find the minimizer of the following LP:
minimize 7x1 9x2
subject to 3x1 + 4x2 5, 5x1 + 3x2 4, 4x1 5x2 3, 4x1 3x2 5.
Do this by drawing the feasible set and check corner
Math 171A: Linear Programming
Instructor: Philip E. Gill
Winter Quarter 2015
Homework Assignment #3
Due Friday January 30, 2015
Remember that the first midterm exam will be held in class on Wednesday, January 28.
As the first midterm is before the deadlin
Math 171A: Linear Programming
Instructor: Philip E. Gill
Winter Quarter 2015
Homework Assignment #1
Due Friday January 16, 2015
I know that you are all aware of the importance of doing the homework assignments. This
is the best way to keep up with the cla
Math 171A: Mathematical Programming
Instructor: Philip E. Gill
Winter Quarter 2009
Homework Assignment #2 Due Friday January 23, 2009 The starred exercises require the use of Matlab. Remember that it is necessary to do all the Matlab assignments to obtain
Math 171A Homework Assignment # 2
Due Date: January 23, 2014
1. (10 points) Consider the linear system Ax = b where
4
3
3
6
2 3
3
6
6
0 ,
A= 1
b= 1
2
1
0
1
6
3
0
5
.
Determine the rank of A and check if Ax = b is compatible. If yes, express all the
Math 171A: Linear Programming
Instructor: Philip E. Gill
Winter Quarter 2015
Homework Assignment #2
Due Friday January 23rd, 2015
The starred exercises require the use of Matlab. Remember that it is necessary to do all
the Matlab assignments to obtain cre
Math 171A: Mathematical Programming
Instructor: Philip E. Gill
Winter Quarter 2009
Homework Assignment #1 Due Friday January 16, 2009 I know that you are all aware of the importance of doing the homework assignments. This is the best way to keep up with t
Solution of Linear Programming Problems with Matlab
Notation
The transposition operation is denoted by a superscript T (apostrophe in Matlab),
1
T
[1, 2, 3] = 2 ,
3
T
1
2 = [1, 2, 3],
3
1 2 3
4 5 6
T
1 4
= 2 5 .
3 6
Given two (row or column) vectors a
Math 171A: Linear Programming
Instructor: Philip E. Gill
Winter Quarter 2015
Homework Assignment #4
Due Friday February 6, 2015
Starred exercises require the use of Matlab.
Exercise 4.1. Suppose that the constant vector c is such that cTp 0 for all p such
Math 171A: Linear Programming
Instructor: Philip E. Gill
Winter Quarter 2015
Homework Assignment #3
Due Friday January 30, 2015
Remember that the first midterm exam will be held in class on Wednesday, January 28.
Starred exercises require the use of Matla
Math 171A LINEAR PROGRAMMING Class Notes
c 1998. Philip E. Gill, Walter Murray and Margaret H. Wright Department of Mathematics University of California, San Diego, La Jolla, CA 92093-0112. January 2007
Contents
1 Background 1.1. Denitions and Operations
Recap: column rank
Math 171A: Linear Programming
Lecture 6
What is the smallest number of linearly independent columns of A
(say r ) that completely define range(A)?
Gather r independent columns so that range(B) = range(A).
Full-Rank Systems of Linear Equ
Math 171A: Linear Programming
Lecture 3
Geometry of the Feasible Region
Philip E. Gill
For the Matlab homework, please use the Matlab diary feature.
>
>
>
>
>
diary mysession.txt
.
.
.
diary off
c 2017
http:/ccom.ucsd.edu/~peg/math171a
Edit the file myses
Recap
Math 171A: Linear Programming
Lecture 2
Properties of Linear Constraints
The lecture slides and homework are posted on the class web-page.
http:/ccom.ucsd.edu/~peg/math171a
Access to course materials requires a class account and password.
Philip E.
Recap: LP with equality constraints
Math 171A: Linear Programming
Linear programming with equality constraints:
Lecture 9
ELP
minimize
n
`(x) = c Tx
subject to
Ax = b
xR
Optimality Conditions for Equality
Constraints
The feasible region is
Philip E. Gill
Recap: The graphical method
Math 171A: Linear Programming
1. Graph the feasible region
Lecture 5
Two Methods for Toy Linear Programs
2. Find a corner point of F and a level curve cfw_x : c Tx = z0 such
that
(a) z0 is as small as possible; and
Philip E. G
Recap: Simplex method for LPs in standard form
Math 171A: Linear Programming
Lecture 22 The Simplex Method for Standard Form
Philip E. Gill
c 2011
minimize n
xR
c Tx Ax = b , x 0
simple bounds
subject to
equality constraints
We apply "mixed-constraint" si
Recap: LP formulations
Math 171A: Linear Programming
Problems considered so far: ELP minimize n
xR
Lecture 20 LPs with Mixed Constraint Types
Philip E. Gill
c 2011
c Tx Ax = b c Tx Ax b
subject to LP minimize
x
http:/ccom.ucsd.edu/~peg/math171a
subject to
Recap: Linear programs in standard form
Math 171A: Linear Programming
Lecture 21 Linear Programs in Standard Form
Philip E. Gill
c 2011
minimize n
xR
c Tx Ax = b , x 0
simple bounds
subject to
equality constraints
The matrix A is m n with shape A = We app
Recap: Choice of the generic form
Math 171A: Linear Programming
minimize d Tw n
w R
subject to Gw f ,
w 0
Lecture 23 Linear Programming Duality
Convert to all-inequality form if m > n, i.e.,
G= Philip E. Gill
c 2011
http:/ccom.ucsd.edu/~peg/math171a
Conve
Class Announcements
Math 171A: Linear Programming
Lecture 8
1
The midterm will be held in class on Wednesday
(February 1st).
2
The midterm is based on material covered in Lectures 17
(Homework Assignments #1 and #2).
3
Bring a UCSD ID, a Blue Book (and a
Recap: Optimality conditions for ELP
Math 171A: Linear Programming
Lecture 10
Feasible Directions for Inequality
Constraints
Linear programming with equality constraints:
ELP
minimize
n
c Tx
subject to
Ax = b
xR
A point x is optimal if and only if
Philip
Math 171A: Linear Programming
So far, we have focused on compatible equations Ax = b.
i.e., equations Ax = b with b range(A).
Lecture 7
Properties of Incompatible Equations
Philip E. Gill
Question
Given A Rmn , how do we characterize vectors b Rm such
tha
Overview of Math 171A
Math 171A: Linear Programming
Lecture 1
Overview of the Class:
Introduction to Optimization
Philip E. Gill
c 2017
http:/ccom.ucsd.edu/~peg/math171a
TAs: Yesheng Huang (AP&M 5412) & Chao Fan (AP&M 5412).
The grade is based on homework