EE 127 / EE 227AT
L. El Ghaoui
1/31/17
Homework #1 - Solutions
Solution 1
1. We have
kxk22 =
n
X
x2i n max x2i = n kxk2 .
i
i=1
Also, kxk
p
x21 + . . . + x2n = kxk2 .
The inequality kxk2 kxk1 is obtained after squaring both sides, and checking that
n
X
i
EE 127 / EE 227AT
L. El Ghaoui
4/?/17
Quiz #2 Practice Problems
Name:
Student ID:
Quiz details
The quiz lasts for 1h30. Notes are not allowed, except for a single two-sided cheat sheet.
There are 8 points. Questions are arranged in order of increasing dif
EE127A / EE 227AT
L. El Ghaoui
2/16/2017
Quiz 1: Solutions
1. Consider the matrix
1 4 6
A = 2 2 6 .
2
4
3
(a) Show that the columns of A = [a1 , a2 , a3 ] are mutually orthogonal, that is, aTi aj =
0 when i 6= j.
(b) Show that we can write A = DB, with B
EE127 - EE227 AT
L. El Ghaoui
3/9/2017
Midterm
NAME:
SID:
The exam lasts 1.5 hours. The maximum number of points is 30. Notes are not allowed
except for a two-sided cheat sheet of regular format.
This booklet is 9 pages total, with extra blank spaces allo
EE 127 / EE 227AT
L. El Ghaoui
HW #6 Solutions
Solution 1 (Convex Problem without strong Duality)
1. The objective is convex. The constraints are jointly convex in x, y because x2 /y is
simply the perspective of f (z) = z 2 . Since the only feasible x is
EE 127 / EE 227AT
L. El Ghaoui
3/2/17
Homework #4 Solutions
Solution 1
1. For p1 , we have the LP formulation
p1
= min t +
x,t,z
n
X
zi :
i=1
zi xi zi , i = 1, . . . , n
t (Ax y)i t, i = 1, . . . , m.
2. Likewise, for p2 , we obtain the convex QP
p2 = min
Optimization Models
EE 127 / EE 227AT
Johannes O. Royset
EECS department
UC Berkeley
Spring 2016
Sp16
1 / 18
LECTURE 4 (Chapter 4, minus 4.4.6)
Symmetric Matrices
Whoever think algebra is a trick in
obtaining unknowns has thought in
vain. No attention sho
Optimization Models
EE 127 / EE 227AT
Johannes O. Royset
EECS department
UC Berkeley
Spring 2016
Sp16
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LECTURE 1
Introduction
Because the shape of the whole
universe is most perfect and, in
fact, designed by the wisest
creator, nothing in all of the
EE 127/227AT Spring 2016 Discussion 1
1
Frobenius Norm
The Frobenius norm for a matrix A Rmn is dened as
m
A
F
n
a2
ij
=
i=1 j=1
1. Show that A
2
F
= Tr(AT A).
2. Let A, B Rmn with R(A) R(B). Show that
A+B
2
F
= A
1
2
F
+ B
2
F
2
Ane Sets and Projections
Optimization Models
EE 127 / EE 227AT
Johannes O. Royset
EECS department
UC Berkeley
Spring 2016
Sp16
1 / 36
LECTURE 3 (Chapter 3, minus 3.7)
Matrices and Linear Maps
The Matrix is everywhere. It is all
around us.
Morpheus
Sp16
2 / 36
Outline
1
Introducti
Optimization Models
EE 127 / EE 227AT
Johannes O. Royset
EECS department
UC Berkeley
Spring 2016
Sp16
1 / 27
LECTURE 5 (Chapter 5, minus 5.3.3)
Singular Value Decomposition
The license plate of Gene Golub
(19322007).
Sp16
2 / 27
Outline
1
The singular val
LECTURE 4
Optimization Models
EECS 127 / EE 227AT
Symmetric Matrices
Laurent El Ghaoui
Whoever think algebra is a trick in
obtaining unknowns has thought in
vain. No attention should be paid
to the fact that algebra and
geometry are different in
appearanc
LECTURE 15
Optimization Models
EECS 127 / EE 227AT
Geometric Programs
Laurent El Ghaoui
There is geometry in the humming
of the strings, there is music in the
spacing of the spheres.
EECS department
UC Berkeley
Pythagoras of Samos
Spring 2017
Sp17
1 / 15
LECTURE 16
Optimization Models
EECS 127 / EE 227AT
Duality
Laurent El Ghaoui
All truth passes through three
stages: First, it is ridiculed;
Second, it is violently opposed;
Third, it is accepted as self-evident.
EECS department
UC Berkeley
Spring 2017
Art
EE 127 / EE 227AT
L. El Ghaoui
1/27/15
Homework Assignment #1
Due date: 1/31/17, in class. Please LATEX your homework solution and submit the printout.
Exercise 1 (Norm inequalities)
1. Show that the following inequalities hold for any vector x:
1
kxk2 k
EE 127 / EE 227AT
L. El Ghaoui
3/2/17
Homework Assignment #4
Due date: 3/16/17 at 11:59 PM on Gradescope. Please LATEX your homework solution.
Exercise 1 (Formulating problems as LPs or QPs)
Formulate the problem
.
pj = min fj (x),
x
for different functio
EE 127 / EE 227AT
L. El Ghaoui
2/16/17
Homework Assignment #3
Due date: 3/2/17, in class. Please LATEX your homework solution and submit the printout.
This homework will require programming solutions to basic optimization problems. To do
so, download a so
EE 127 / EE 227AT
L. El Ghaoui
1/31/17
Homework Assignment #2
Due date: 2/14/17, in class. Please LATEX your homework solution and submit the printout.
Exercise 1 (A lower bound on the rank) Let A Sn+ be a symmetric, positive semidefinite matrix.
1. Show
EE 127 / EE 227AT
L. El Ghaoui
2/16/17
Homework Solutions #3
Exercise 1 (Convexity of functions)
1. For x, y both positive scalars, show that
yex/y = max (x + y) y ln .
>0
Use the above result to prove that the function f defined as
x/y
ye
if x > 0, y >
EE 127 / EE 227AT
L. El Ghaoui
1/31/17
Homework Solution #2
Exercise 1 (A lower bound on the rank) Let A Sn+ be a symmetric, positive semidefinite matrix.
1. Show that the trace, trace A, and the Frobenius norm, kAkF , depend only on its
eigenvalues, and
EE 127 / EE 227AT
L. El Ghaoui
1/30/15
Discussion #2
Exercise 1 (Frobenius norm and random inputs) Let A Rm,n be a matrix. Assume
that u Rn is a vector-valued random variable, with zero mean and covariance matrix In .
That is, Ecfw_u = 0, and Ecfw_uu> =
EE 127 / EE 227AT
L. El Ghaoui
1/27/15
Discussion #1
Exercise 1 (Derivatives of composite functions)
1. Let f : Rm Rk and g : Rn Rm be two maps. Let h : Rn Rk be the composite
map h = f g, with values h(x) = f (g(x) for x Rn . Show that the derivatives of
EE 127/227A Discussion, Week of 2/6/17
February 8, 2017
1
HomeworkWhat We Want
When we grade your homework, we are looking for the following:
Mathematically precise answers. (Use terms correctly! If you are in any
way unsure of what youre writing, Google
EE 127 / EE 227AT
L. El Ghaoui
Discussion 6 (Week of Monday, 2/27/17)
Optimization Theory
Exercise 1 (Monotonicity and locality). Consider the optimization problems (no assumption
of convexity here):
.
p1 = min f0 (x)
xX1
.
p2 = min f0 (x)
xX2
.
p13 =
min
Optimization Models
EE 127 / EE 227AT
Johannes O. Royset
EECS department
UC Berkeley
Spring 2016
Sp16
1 / 46
LECTURE 2 (Chapter 2)
Vectors and Functions
Mathematicians are like
Frenchmen: whatever you say to
them, they translate into their own
language, a