ESE518 Spring 2014: Optimization Methods in Control
http:/www.ese.wustl.edu/~hgonzale/ese518
Lecturer: Humberto Gonzalez
Scribe: Sensen Liu
Lecture 10 February 17
10.1
Convex Analysis (contd)
Theorem 10.1. Let U Rn , and f : U R a dierentiable function (i

ESE518 Spring 2014: Optimization Methods in Control
Lecturer: Humberto Gonzalez
Scribe: Han Liu
http:/www.ese.wustl.edu/~hgonzale/ese518
Lecture 9 February 12
9.1
Convex Topology (contd)
Example 9.1. Let cfw_xi k be anely independent, let S = conv cfw_xi

ESE518 Spring 2014: Optimization Methods in Control
Lecturer: Humberto Gonzalez
Scribe: Han Liu
http:/www.ese.wustl.edu/~hgonzale/ese518
Lecture 8 February 10
8.1
Convex Topology
Throughout this section we will assume that V is a vector space.
Denition 8.

ESE518 Spring 2014: Optimization Methods in Control
http:/www.ese.wustl.edu/~hgonzale/ese518
Lecturer: Humberto Gonzalez
Scribe: Ignacio de Erausquin
Lecture 6 February 3
6.1
Recap: Inner Product Spaces (contd)
Denition 6.1. Given two subspaces of V , say

ESE518 Spring 2014: Optimization Methods in Control
http:/www.ese.wustl.edu/~hgonzale/ese518
Lecturer: Humberto Gonzalez
Scribe: Bo Li
Lecture 5 January 29
5.1
Recap: Normed Spaces (contd)
Denition 5.1. Let (U, U ) and (V, V ) be normed spaces, and let f

ESE518 Fall 2013: Homework 5 Solution
Humberto Gonzalez
February 28, 2014
Problem 1
Let H1 = x | aT x = b1 and H2 = x | aT x = b2 . Note that the vector a is orthogonal to both hypera
planes, and that given x H1 the closest point in H2 to x is y = x + a f

ESE518 Fall 2013: Homework 4 Solution
Humberto Gonzalez
February 19, 2014
Problem 1
Note that, based on the notation of this problem, U Rmr , Rrr , and V Rrn .
(1) If r = n, then V, Rnn , and U Rmn .
Hence, AT A = V 2 V T is the product of fullrank matric

ESE518 Fall 2013: Homework 1 Solution
Humberto Gonzalez
January 31, 2014
Problem 1
(a) First, assume that f has a left inverse, say gL . Then, given a1 , a2 A s.t. f (a1 ) = f (a2 ), since
(gL f )(a) = a for each a A, we get that a1 = a2 , which means tha

ESE518 Spring 2014: Optimization Methods in Control
http:/www.ese.wustl.edu/~hgonzale/ese518
Lecturer: Humberto Gonzalez
Scribe: Humberto Gonzalez
Lecture 1 January 13
1.1
Motivation
Research in engineering is mostly driven by results in theoretical mathe

ESE518 Fall 2013: Homework 3 Solution
Humberto Gonzalez
February 12, 2014
Problem 1
(a) Let x Rn , and let k cfw_1, . . . , n be an index such that xk = x
,
i.e.
k arg max |xk | | k cfw_1, . . . , n .
(1)
Then:
n
x
= xk
n
x
1
|xk | = x 1 ,
|xk |
=
and
(

ESE518 Fall 2013: Homework 2 Solution
Humberto Gonzalez
February 1, 2014
Problem 1
n
n
i=1
(a) Suppose that cfw_ui 1 is l.d. Then there exists Rn , = 0, such that
n
0=f
i ui = 0, and:
n
i ui
i=1
=
i f (ui ),
(1)
i=1
n
n
but this is a contradiction since c

ESE518 Spring 2014: Optimization Methods in Control
http:/www.ese.wustl.edu/~hgonzale/ese518
Lecturer: Humberto Gonzalez
Scribe: Runxin He
Lecture 2 January 15
2.1
Recap: Vector Spaces (contd)
Denition 2.1. We say that (V, R) is a vector space (or VS) con

ESE518 Spring 2014: Optimization Methods in Control
http:/www.ese.wustl.edu/~hgonzale/ese518
Lecturer: Humberto Gonzalez
Scribe: Runxin He
Lecture 3 January 22
3.1
Recap: Linear Maps (contd)
Denition 3.1. rank(f ) = dim R(f ) and nullity(f ) = dim N (f )

ESE518 Spring 2014: Optimization Methods in Control
Lecturer: Humberto Gonzalez
Scribe: Ignacio de Erausquin
http:/www.ese.wustl.edu/~hgonzale/ese518
Lecture 7 February 5
7.1
Recap: Inner Product Spaces (contd)
Proof of Lemma 6.12. This is just a sketch o

ESE518 Spring 2014: Optimization Methods in Control
http:/www.ese.wustl.edu/~hgonzale/ese518
Lecturer: Humberto Gonzalez
Scribe: Ignacio de Erausquin
Lecture 6 February 3
6.1
Recap: Inner Product Spaces (contd)
Denition 6.1. Given two subspaces of V , say

ESE518 Spring 2014: Optimization Methods in Control
Lecturer: Humberto Gonzalez
Scribe: Bo Li
http:/www.ese.wustl.edu/~hgonzale/ese518
Lecture 4 January 27
4.1
Recap: Normed Spaces
Denition 4.1. Let (V, R) be a vector space. We say that (V, ) is a normed