Math 171A Homework Assignment # 2
Due Date: January 23, 2014
1. (10 points) Consider the linear system Ax = b where
3
3
6
4
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
Section 1.3
4.
6.
.
10.
[
]
[
]
[
]
[ ].
12. Yes, b is a linear combination of a1, a2, and a3.
14. Yes, b is a linear combination of the columns of A.
16. h = 2.
18. Non-integer weights are acceptable, of course, but some simple choices are 0v1 + 0v2 = 0,
Section 1.1
4. The line equations are:
.
The augmented matrix is:
[
].
To eliminate x1 term in second equation, add (-3) times row 1 to row 2.
[
]
Multiply row 2 by -1/8 in order to obtain 1.
[
]
The augmented matrix is now in triangular form. To interpre
Math 171B: Numerical Optimization: Nonlinear Problems
Instructor: Michael Holst
Spring Quarter 2015
Solutions for Homework Assignment #1
Exercise 1.1. If x is an eigenvector of A, show that x is also an eigenvector for any 6= 0. What is the
associated eig
Math 171B: Numerical Optimization: Nonlinear Problems
Instructor: Michael Holst
Spring Quarter 2015
Solutions for Homework Assignment #2
Exercise 2.1. Sketch F (x) = 1/x a for a = 2. Then derive a Newton iteration for computing the
reciprocal of a positiv
Math 171B: Numerical Optimization: Nonlinear Problems
Instructor: Michael Holst
Spring Quarter 2015
Solutions for Homework Assignment #3
Exercise 3.1. Let H be a symmetric matrix with spectral decomposition H = V DV T .
(a) Show that an eigenvector v asso
Math 171B: Numerical Optimization: Nonlinear Problems
Instructor: Michael Holst
Spring Quarter 2015
Solutions for Homework Assignment #4
Exercise 4.1. Let f (x) denote a convex continuously differentiable function. Show that if a stationary
point x exists
Math 171A Homework Assignment # 7
Instructor: Jiawang Nie
Due Date: March 14, 2014
Ax = b, x 0 with
1
]
[
]
0
1 1
1
,b =
, c=
2 .
3 2
2
0
1. (10 points) Consider the LP: min
[
A=
1
1
1
1
cT x
s.t.
(a) Use any method of your choice to nd a vertex for th
Tentative Schedule of Math 171A, Winter 2014
Jiawang Nie
The textbook is Linear Programming (by Philip Gill et al., 1998). To obtain an electronic copy,
send a request email to the instructor.
Week 1:
01/06: Introduction to Optimization
01/08: Review on
Math 171A Practice Midterm II
Notes: 1) For computational problems, no credit will be given for unsupported answers
gotten directly from a calculator. 2) For proof problems, no credit will be given for
wrong reasons.
1. Consider the LP of minimizing
2
3
A
What is optimization?
Lecture 1
Introduction to Optimization
Websters dictionary:
UCSD Math 171A: Numerical Optimization
Optimization the process or method for making something
(design, system, decision) as fully perfect, functional or eective as
possible
Math 171A Homework Assignment # 4
Due Date: February 14, 2014
1. (10 points) Consider the feasible set F dened 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, and nd all the corner poin
Math 171A Homework Assignment # 3
Due Date: January 31, 2014
1. (10 points) Let F = cfw_x R2 : 1 x1 x2 1, x2 0. Describe
the sets of vectors c such that:
(a) cT x is bounded from below but unbounded from above;
(b) cT x is unbounded from below but bounded
Math 171A Homework Assignment # 1
Due Date: Thursday, January 16, 2014
1. (10 points) Find the minimizer of the following LP:
cfw_
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 co
Section 1.9
4. [
]
8. [
]
10. [
]
12. The transformation T in Exercise 10 maps e1 into e2 and maps e2 into e1. A
counterclockwise rotation about the origin through /2 radians also maps e1 into e2 and maps
e2 into e1. Since a linear transformation is compl
Section 2.1
[
2.
], 2C 3E is not defined,
[
], EC is not defined.
[
4.
(
)
],
[
].
[
6. a.
],
[
],
( )
( )]
( )
[
b.
[
].
[
].
8. B has 5 rows.
[
10.
].
12. By inspection of A, a suitable column for B is any multiple of (2, 1). For example:
[
].
16. a. Tr
Section 1.7
[ ]
2. Let
[ ]
[
[
]
]
[
[
]
]. Consider the equation:
.
The augmented matrix is:
[
].
Interchange row 1 and row 3, and add 2 times row 1 to (-3) times row 2:
[
].
It is clear that there are 3 basic variables and no free variables. So the equa