Math 408
Homework Set 2
This homework set will focus on the linear least squares problem
LLS
min
n
xR
1
Ax b
2
2
2
,
where A Rmn and b Rm .
(1) Listed below are two functions. In each case write the p
Math 408
Homework Set 2
This homework set will focus on the linear least squares problem
LLS
min
n
xR
1
Ax b
2
2
2
,
where A Rmn and b Rm .
(1) Listed below are two functions. In each case write the p
Math 408
Homework Set 2
This homework set will focus on the linear least squares problem
1
LLS
min Ax b 2 ,
2
n 2
xR
where A Rmn and b Rm .
(1) Listed below are two functions. In each case write the p
MATH 408
MIDTERM GUIDE SOLUTIONS
OUTLINE
Sample Questions
(I) Linear Least Squares
Question 1:
Let A Rmn and b Rm , and consider the linear least squares problem
LLS
min
1
Ax b
2
2
2
.
a. Show that th
Math 408
Homework Set 2
This homework set will focus on the linear least squares problem
1
LLS
minn kAx bk22 ,
xR 2
mn
m
where A R
and b R .
(1) Listed below are two functions. In each case write the
(page 1 of 5)
(1) Consider a function f: IR" H R. State the denitions of the following properties
(a) (3 Points) f is coercive.
um +(X) : +00
We 100
(b) (3 points) I 6 IR" is a local minimizer of f.
L
MATH 408
FINAL EXAM
March 5, 2012
SAMPLE
Partial Solutions to Sample Questions (in progress)
See the sample questions for the midterm exam, but also consider the following questions. Obviously, a fina
AMATH/MATH 516
FIRST HOMEWORK SET SOLUTIONS
The purpose of this problem set is to have you brush up and further develop your multi-variable calculus and linear
algebra skills. The problem set will be
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2014/9/17
page 117
Chapter 7
Convex Functions
7.1 Definition and
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2014/9/17
page 169
Chapter 9
Optimization over a
Convex Set
9.1 S
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2014/9/17
page 147
Chapter 8
Convex Optimization
8.1 Definition
A
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2014/9/17
page 97
Chapter 6
Convex Sets
In this chapter we begin
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2014/9/17
page 83
Chapter 5
Newtons Method
5.1 Pure Newtons Metho
1. Review of Multi-variable Calculus
Throughout this course we will be working with the vector space Rn . For this reason we
begin with a brief review of its metric space properties
Definition 1.1 (Ve
Math 408
Homework Set 1
Linear Algebra Review Problems
(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) Red
MATH 408
FINAL EXAM
SOLUTIONS TO SAMPLE QUESTIONS
Sample Questions
Question 1: Theory Question
1. State the rst- and second-order conditions for optimality for the following two problems:
(a) Linear l
Math 408
Homework Set 5
Solutions
(1) Show that the functions
f (x1 , x2 ) = x2 + x3 ,
1
2
and g(x1 , x2 ) = x2 + x4
1
2
both have a critical point at (x1 , x2 ) = (0, 0) and that their associated Hes
Math 408
Homework Set 6
(1) Use the delta method (as we did in class for the Linear Least Squares function) to compute
the gradient and the Hessian of the following functions.
(a) f (x) := 1 Ax b 2 ,
Math 408
Solutions
This homework set will focus on the following two optimization problems:
1
min xT Hx + g T x
n 2
xR
Q
and
LLS
min
n
xR
1
Ax b 2 ,
2
2
where H Rnn is symmetric, g Rn , A Rnk and b Rn
Math 408
Homework Set 3 Solutions
This homework set will focus on the optimization problem
1
Q
minn xT Hx + g T x ,
xR 2
nn
n
where H R
is symmetric and g R .
(1) Each of the following functions can b
Math 408
Homework Set 5
(1) Show that the functions
f (x1 , x2 ) = x2 + x3 ,
1
2
(2)
(3)
(4)
(5)
(6)
(7)
(8)
and g(x1 , x2 ) = x2 + x4
1
2
both have a critical point at (x1 , x2 ) = (0, 0) (i.e. f (x1
Math 408
This homework set will focus on the following two optimization problems:
1
Q
min xT Hx + g T x
xRn 2
and
1
LLS
min Ax b 2 ,
2
n 2
xR
where H Rnn is symmetric, g Rn , A Rnk and b Rn . For easy
Nonlinear Optimization
James V. Burke
University of Washington
Contents
Chapter 1.
Introduction
5
Chapter 2. Review of Matrices and Block Structures
1. Rows and Columns
2. Matrix Multiplication
3. Blo
CHAPTER 2
Review of Matrices and Block Structures
Numerical linear algebra lies at the heart of modern scientic computing and computational science. Today
it is not uncommon to perform numerical compu
1. Review of Multi-variable Calculus
Throughout this course we will be working with the vector space Rn . For this reason we
begin with a brief review of its metric space properties
Denition 1.1 (Vect
MATH 408 QUIZ
NAME (Please print):
There are 2 problems. Stop now and make sure you have both problems. If you do not have them
both, then request a new quiz. The rst problem is worth 30 points and th
MATH 408
MIDTERM GUIDE SOLUTIONS
OUTLINE
Sample Questions
(I) Linear Least Squares
Question 1:
Let A Rmn and b Rm , and consider the linear least squares problem
LLS
1
min kAx bk22 .
2
a. Show that th
MATH 408
MIDTERM EXAM
OUTLINE
The midterm exam will consist of two parts: (I) Linear Least Squares and (II) Quadratic Optimization.
In each part, the first question concerns definitions, theorems, and