Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
EE 680/M 520, Spring 2002
Exam 2: Session 26
Solutions
75 mins.; Total 50 pts.
1. (12 pts.) Consider the optimization problem
minimize
subject to
f (x)
x ,
where Rn . Suppose we apply the penalty method to the above problem, which involves solving an asso
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
EE/M 520, Spring 2003
Exam 2: Session 26
Solutions
75 mins.; Total 50 pts.
1. (15 pts.) Consider the optimization problem
minimize
subject to
f (x)
x ,
where f : R2 R, f C 1 , and = [1, 1]2 = cfw_x : 1 xi 1, i = 1, 2. Consider the
projected steepest desce
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2009
Funwork #3
Solutions
1. The Taylor series expansion of a function f(x) about x=x0 is
f ( k ) ( x0 )
f ( x0 )
f ( x0 )
f ( x) =
( x x0 ) k = f ( x 0 ) + f ( x0 )( x x0 ) +
( x x0 ) 2 +
( x x 0 ) 3 + .
k!
2!
3!
k =0
Since f ( x) = cos x
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
1.
Consder the following linear program:
minimze z = x1 + x2 + x3 ,
x1 + 2 x2 2 x3 0,
subject to x1 + x3 1,
x1 , x2 , x3 0.
a. Solve this problem for = 1 by using twophase simplex method. (10 pts)
b. Which vales of leads to an unbounded dual problem? Mo
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
The Split Bregman Method for L1Regularized
Problems
Tom Goldstein
May 22, 2008
Some Common L1 Regularized Problems
TV Denoising: min u
u
BV
+
DeBlurring/Deconvolution: min u
u
uf
2
BV
+
Basis Pursuit/Compressed Sensing MRI: min u
u
2
2
Ku f
2
BV
+
2
2
F
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2011
FunWork #5
Due on April 22
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer. O
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
Funwork #4 Solution
1
Exercise 12.10 from TEXT on page 240
y (t) = sin(t + )
a. We can represent the problem as a system of p equations as follows.
y1
y2
=
=
.
.
.
sin(t1 + )
sin(t2 + )
yi
=
.
.
.
=
sin(ti + )
yp
Since ti +
2
2 , i
sin(tp + )
= 1, ., p,
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2011
FunWork #4
Due on March 25
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer. O
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2009
Funwork #4
Due on February 23
1. For the function
f (x1 , x2 ) = 4(x1 2)2 + (x2 3)2 ,
(a) Obtain the sequence of the rst six points using the steepest descent method and locate
these points on the level sets of f . The initial guess is
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2009
Funwork #4
Solutions
1. (a) Steepest Descent Algorithm
The steepest descent algorithm is as follows:
For the quadratic objective function,
1
1 8 0
T 16
f ( x) = x T Q x x T b + c = x T
x x 6 + 25,
2
2 0 2
(0 )
x = 0,
x ( k +1) = x (
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
FunWork 3
ECE 580, Purdue University
March 5, 2011
1
ECE580
1
FunWork: 3
In this assignment, we are confronted with the Peaks function as a testing ground for several
minimization algorithms. There are four NewtonBased methods being tested, and there is
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2011
FunWork #3
Due on March 09
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer. O
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
Homework #3
Due on Dec. 27
1. Suppose that we perform an experiment to calculate the gravitational constant g as follows.
We drop a ball from a certain height and measure its distance from the original point at certain
time instants. The results of the ex
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
Absolute Funwork #2 from ECE 580
February 28, 2011
1
Question 1
*Before I go, I mostly solved this problem with matlab. Thus, it will be a little or no formula
explanation presented.
a) Using the matlab, I can plot f (x) versus x as follows:
Figure 1: Pro
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2011
FunWork #2
Due on February 16
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer.
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2009
Funwork Set #3
Due on February 13
1. Use the Taylor series expansion to approximate f (x) = cos x about x0 = 0. Plot f , the
zerothorder, the secondorder, and the fourthorder approximations on the same plot, where
6 x 6. The range f
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2009
Funwork #2
Due on February 2, 2009
1. Consider the problem of solving a jigsaw puzzle that consists of N pieces. Is this problem
P, nonP, or NP? Justify your answer.
2. Exercise 5.9 from TEXT on page 75. Verify your calculations using
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2009
Funwork #2
Solutions
1. Consider the problem of solving a jigsaw puzzle that consists of N pieces. Is this
problem P, nonP, or NP? Justify your answer.
Answer: This problem comes from the article MillionDollar Minesweeper by Ian
Stew
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
First Semester 2012
2104493 Computational Methods in Financial Engineering
2104662 Optimization for Financial Engineering
Homework 2
Homework
due on 10 Jul 2012 for 2104662 and 12 Jul 2012 for 2104493
Question 1 [10 points]
For each of the following funct
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
Homework #2
Due on Dec. 18
1. Consider Rosenbrocks Function:
2
where
= [1 , 2 ] .
2
f(x) = 100(2 1 ) + (1 1 )
2
(This function is known as a nasty function and often used as a
benchmark for testing algorithms.) This function is also known as the banana f
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2011
FunWork #1
Due on February 2
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer.
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2009
Linear Algebra Review
Due on January 26, 2009
1. Investigate the rank of the following matrix for dierent values of the parameter ,
1 1 2
A = 2 1 5 .
2. Let
1 10 6 1
1 2 1 3 2
2 1 3 0 1
A=
3
1
2
33
1
2
3
11
Find the rank of the abo
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
ECE 580
Spring 2009
Funwork #1
Solutions
1.
For the matrix,
1
2
2 1
A=
3 1
12
1 3 2
3
2
3
0 1
,
3 3
11
nd its rank by rst transforming the matrix by means of the row elementary operations
into an upper triangular form.
Find the rank of the following ma
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
Homework #1
Due on Dec. 11
1. Compute the linear
at the point
l(1 , 2 ), and quadratic q(1 , 2 ), approximations of the function
(0) = [1 1]
3
22
f = f(1 , 2 ) = 1 + 1 2 1 2
2.
(a) Find Df(x) of
f(x) = [
(b) Find the Hessian of
2
15
] [ ] +
3
23
1
23
1
Korea Advanced Institute of Science and Technology
ECON 781

Spring 2012
Applications of Optimization Theory
Spring 2004
Exam 1: 2004/4/14
Total 100 pts
1. (30 pts.) Consider the problem
minimize f(x)
subject to x
2
T
where f C and = cfw_x =[x1, x2] : x1 x22 +2 x2, x2 0. Consider the point x*= 0 = [0,
T
0] .
T
a. Suppose f(x*