EE 127 / EE 227AT
L. El Ghaoui
1/20/15
Homework Assignment #1
A
Due date: 2/3/15, in class. Please L TEX your homework solution and submit the printout.
Exercise 1 (Some simple problems.) Show the following results.
1. For any A R, and > 0:
(A+ )2
,
1 (A,
EE 127 / EE 227AT
L. El Ghaoui
HW #6 Solutions
Exercise 1 (Transaction costs and market impact) We consider the following portfolio optimization problem
max r x x Cx c T (x x0 ) : x 0, x X ,
x
(1)
where C is the empirical covariance matrix, > 0 is a risk
EE 127 / EE 227AT
L. El Ghaoui
HW #1 Solutions
Exercise 1 (Some simple problems.) Show the following results.
1. For any A R, and > 0:
2 (A+ )2
,
1 (A, ) := max A =
0
2
2
.
with unique optimal point = A+ /, where A+ = maxcfw_A, 0.
2. For A, B R, and > 0,
EE 127 / EE 227AT
L. El Ghaoui
2/3/15
Homework Assignment #2
A
Due date: 2/17/15, in class. Please L TEX your homework solution and submit the printout.
Exercise 1 (Projections and PCA Computation) The dataset for this problem consists of the votes of n =
EE 127 / EE 227AT
L. El Ghaoui
HW #4 Solutions
Exercise 1 (Monotonicity and locality) Consider the optimization problems (no assumption of convexity here)
.
p = min f0 (x)
1
xX1
.
p2 = min f0 (x)
xX2
.
p13 =
min f0 (x)
xX1 X3
.
p =
min f0 (x),
23
xX2 X3
w
EE 127 / EE 227AT
L. El Ghaoui
HW #3 Solutions
Exercise 1 (Bounds on a quadratic-fractional function) We consider the function f :
.
Rn R, with values on its domain X = cfw_x : x 2 1 given by
f (x) =
x Bx
.
1 x Ax
Here, A, B Rn,n are given, with A, B symm
EE 127 / EE 227AT
L. El Ghaoui
3/3/15
Homework Assignment #4
A
Due date: 3/17/15, in class. Please L TEX your homework solution and submit the printout.
Exercise 1 (Monotonicity and locality) Consider the optimization problems (no assumption of convexity
EE 127 / EE 227AT
L. El Ghaoui
17/2/15
Homework Assignment #3
A
Due date: 3/3/15, in class. Please L TEX your homework solution and submit the printout.
Exercise 1 (Bounds on a quadratic-fractional function) We consider the function f :
.
Rn R, with value
EE 127 / EE 227AT
L. El Ghaoui
HW #2 Solutions
Exercise 1 (Projections and PCA Computation) The dataset for this problem consists of the votes of n = 100 Senators in the 2004-2006 US Senate for a total of m = 542
bills. Yay (Yes) votes are represented as
EE 127 / EE 227AT
L. El Ghaoui
3/19/15
Homework Assignment #5
A
Due date: 4/9/15, in class. Please L TEX your homework solution and submit the printout.
Exercise 1 (Auto-regressive process model) We consider a process described by difference equation
y(t
EE 127 / EE 227AT
L. El Ghaoui
4/9/15
Homework Assignment #6
A
Due date: 4/23/15, in class. Please L TEX your homework solution and submit the printout.
Exercise 1 (Transaction costs and market impact) We consider the following portfolio optimization prob
EE 127 / EE 227AT
L. El Ghaoui
HW #5 Solutions
Exercise 1 (Auto-regressive process model) We consider a process described by difference equation
y(t + 2) = 1 (t)y(t + 1) + 2 (t)y(t) + 3 (t)u(t), t = 0, 1, 2, . . .
where the u(t) R is the input, y(t) R the
EE127A
L. El Ghaoui
2/19/15
Quiz 1: Solutions
1. Consider an orthonormal matrix U = [u1 , . . . , un ] Rnn , with uT uj = 0 when i = j,
i
and ui 2 = 1, i = 1, . . . , n. Let Rn , and consider the matrix A = [1 u1 , . . . , n un ]
2
(in words: the columns
EE 127 / EE 227AT
L. El Ghaoui
3/12/15
Midterm: Solutions
1. (10 points) Low-rank matrix completion. We consider a m n data matrix that is only
partially known. We wish to ll in the unknown entries based on the principle that
the completed matrix should b
EE127A
L. El Ghaoui
4/14/15
Quiz 2: Solutions
1. We consider a resource allocation problem of the form
max min rT w
wW
rE
where W := cfw_w Rn : w1 + . . . + wn = 1, and
+
E := cfw_ + Du :
r
u
2
1 .
Here, r Rn and D = diag(1 , . . . , n ) are given, with