ECE 580
Spring 2014
FunWork #3
Due on March 12
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.
Mi
ECE580, Funwork #4
1. (Exercise 12.31) Construct matrices A1 and A2 such that (A1 A2 ) = A A .
2 1
Answer:
1 1
If we set A1 = 0 0 and A2 =
0 0
0.5 0
0.25
and A =
so, A A =
2
2 1
0.5 0
0.25
0.5 0 0
0.5 0 0
1 1
but A =
, then (A1 A2 ) =
1
0.5 0 0
0.5 0 0
0
ECE 580
Spring 2014
FunWork #2
Due on February 19
1. Compute the linear, l(x1 , x2 ), and quadratic, q(x1 , x2 ), approximations of the function
f = f (x1 , x2 ) = x3 + x1 x2 x2 x2 ,
1
1 2
at the point x(0) =
1 1
.
2.
(a) Find Df (x) of
2
1 5
f (x) = x
ECE 580
Basic Optimization Problem
Section 6.1
Motivation, Notation and Vocabulary
Stan Zak
January 13, 2014
ECE 580Basic Optimization Problem p. 1/17
Motivation
Optimizationan act, process, or
methodology of making something (as a
design, system, or deci
1. (10 pts) Consider the function
f(9') = (0%)?ka
Where a, b, and ac are ndimensional vectors.
(i) (5 pts) Find Vf(a:).
(ii) (5 pts) Find the Hessian F(:I;).
cfw_i7 WK : (MM an ~ '4 mm) _ bTJ=(111,+IML-cfw_-v"f[yr/7n).
M); fnpx g (2L17(m[i7+7267)7(
97M]:
ECE 580 ' . Spring 2007
Final Exam
Solutions
1. (10 pts) Given the following function,
331:32): n + $356!.
(i) (3 pts) In What direction does the function f increase most rapidly at the point
2(0) = [ 2 1 F?
(ii) (4 pts) What is the rate of increase of f
ECE 580
Spring 2006
Midterm #2
Solutions
1. Use the least squares method to find the circle
x2 + y 2 = r 2 ,
that comes as close as possible to the three data points:
(x1 , y1 ) = (2, 0),
(x2 , y2 ) = (0, 1),
(x3 , y3 ) = (1, 1).
Answer: There are a numbe
ECE 580
Spring 2009
Funwork #1
Solutions
1.
For the matrix,
1
2
2 1
A=
3 1
1 2
1 3 2
3
2
3
0 1
,
3 3
1 1
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 followi
ECE 580
Spring 2008
Funwork #5
Due on April 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. O-
ECE 580
Spring 2014
FunWork #4
Due on March 28
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.
1.
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
ECE 580
Spring 2008
Funwork #3
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) =
ECE 580
Spring 2008
Funwork #3
Due on March 5
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-c
ECE 580
Spring 2008
Funwork #2
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 x 0 ) +
( x x0 ) 2 +
( x x0 ) 3 +
k!
2!
3!
k =0
Since f ( x) = cos x a
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, non-P, or NP? Justify your answer.
2. Exercise 5.9 from TEXT on page 75. Verify your calculations using
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.
ECE580 Funwork #3
Yukun An
March 11, 2015
The 2D Griewank function is:
f (x1 , x2 ) =
1
1
x2
x1 2 +
x2 2 cos(x1 )cos( ) + 1
4000
4000
2
The contour plot is needed to show the steps of minimizing. So the contour plot of
the 2D Griewank function is shown be
ECE580 Funwork #5
Yukun An
4/24/2015
1. Use a simple method to solve a linear program of Exercise 16.11 on page 373
minimize x1 + x2
subject to
x1 + 2x2 3
2x1 + x2 3
x1 , x2 0
The problem in standard form is:
minimize x1 + x2 + 0x3 + 0x4
subject to
x1 + x
ECE 580: Intro to Algorithms
Stan Zak
School of Electrical and Computer Engineering
Purdue University
zak@purdue.edu
January 10, 2014
1 / 40
Todays Class
Denition of an algorithm
2 / 40
Todays Class
Denition of an algorithm
Central question of the math of
ECE 580
Spring 2013
Midterm #2
Solutions
1. Maximize
1
f = x1 x2 x2 x2 + 3
2
2 1
using the BFGS method. The starting point is x(0) =
Answer: Our objective is a quadratic of the form
0 0
and H 0 = I 2 .
1
1
1 0
+3
x + x
f = x
2
1
0 2
1
= x Qx + x b + c
ECE580 FunWork2
February 19, 2014
Q1) Compute the linear, l(x1 , x2 ), and quadratic, q(x1 , x2 ), approximations of the function
f = f (x1 , x2 ) = x3 + x1 x2 x2 x2
1
1 2
at the point x (0) = 1
1
T
.
Using Theorem 5.8 Taylors Theorem
Assume that a functi
ECE 580
Spring 2014
FunWork #5
Due on April 23 (Wednesday)
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 resultin
ECE 580
Optimization Methods for
Systems and Control
Chapter 11
Quasi-Newton Methods
Stan Zak
February 25, 2013
ECE 580Optimization Methods for Systems and Control p. 1/16
Quasi-Newton Algorithms
x(k+1) = x(k) + k d(k)
d(k) = H k g (k)
k = arg min f x(k)
ECE580 FunWork1
February 5, 2014
1. Theorem 2.1
The system of equations Ax = b has a solution if and only if
rank A = rank[A, b]
(a)
x1 + x2 + 2x3 + x4 = 0
2x1 + 2x2 + 4x3 + 2x4 = 1
According to Theorem 2.1,
[
1 1
A=
2 2
2
4
1
2
]
[ ]
0
b=
1
and
So rank A
ECE 580
Spring 2014
FunWork #1
Due on February 5
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.
ECE 580 Lectures 3 and 4
Basic Optimization Problem
Section 6.2
Denitions and Fundamental Results
Stan Zak
January 15, 2014
ECE 580 Lectures 3 and 4Basic Optimization Problem p. 1/42
Optimization problem statement
minimize
subject to
f (x)
x
ECE 580 Lectu
ECE580, Funwork #3
TAs comments : Since the steepest descent is too simple, here I just skip it. Instead, I showed
Newtons Method for your study.
Q1) Minimize the function
f (x1 , x2 ) = (x2 x1 )4 + 12x1 x2 x1 + x2 + 3
using
1.
2.
3.
4.
5.
Newtons method;
EE 580
Spring 2001
Midterm #2
April 12, (Thursday) 2001
Name:
Student ID #:
Problem Weight Score
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
Total
100
1
1. Suppose that the Kaczmarz algorithm is applied to solve a system of equations represented as
ECE 580
Spring 2007
Midterm #1
Solutions
1. Does the function,
1 2
1
1
x xT
,
f (x) = xT
2
0 1
1
have a minimizer? If it does, then find it; otherwise explain why it does not.
Answer: If a point is a minimizer of f (x) then it satisfies the FONC,
f (x)
ECE 580 Spring 2008
Midterm #1
Solutions
1. Does the function,
f(:c)=:cT[:)2 21]m+ccT[11], $ER2,
have a minimize]: or a maximizer? If it does, then nd it; otherwise explain why it
does not.
Answer: If a point is a minimizer or a maximizer of f (as) then i