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,
ECE 580
Spring 2011
Midterm #2
March 30, (Wednesday)
Name:
Student ID #:
Problem Weight Score
1
15
2
15
3
15
4
15
5
10
6
15
7
15
Total
100
This is a 50 minute duration exam
There are seven questions
Closed books, closed notes, no crib-sheets, no calcul
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 2004
Midterm #2
April 15, (Thursday) 2004
Name: Student ID #: Problem Weight Score
1 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 10 10
Total
This is a 75 minute duration exam Closed books, closed notes No crib-sheets, no calculators There a
ECE 580
Spring 2004
Midterm #2
Solutions
1. The transfer function, (s)/Va (s), modeling of an armature-controlled dc motor with negligibly small armature inductance is (s) 0.5 . = Va (s) K1 s + K2 The corresponding dierential equation is K1 d(t) + K2 (t)
EE 580
Spring 2001
Midterm #1
February 22, (Thursday) 2001 1. For the function
f = f (x1, x2) = x2x2 + x3x1, 1 2
(a) (3 pts.) nd the gradient of f at x =
T
21
;
T
(b) (3 pts.) nd the rate of increase of f at the point x =
T
21
in the direction
d=
43
;
T
(
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-
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-
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 Newton-Based methods being tested, and there is
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-
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-
ECE 580
Spring 2011
Midterm #2
Solutions
1. Minimize
1
f = (x2 + 2x2 ) x1 + x2 + 7
2
21
using the rank two correction (DFP) method. The starting point is x(0) =
00 .
Answer: Our objective is a quadratic of the form
1 1 0
1
+7
x x
x
f=
2
1
02
1
=
x Qx x
EE 580
Spring 2001
Midterm #1
February 22, (Thursday) 2001
Solutions
1. For the function
f = f (x1 , x2 ) = x2 x2 + x3 x1 ,
1
2
(a) (3 pts.) nd the gradient of f at x =
T
21
;
(b) (3 pts.) nd the rate of increase of f at the point x =
d=
T
43
T
in the dir
ECE 580
Spring 2013
FunWork #5
Due on April 19 (Friday)
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 a
A Dynamical System Based Method for Solving Linear
Programs
by
Stanislaw H. Zak
March 4, 2013
1
The Solver
An interesting non-simplex method for solving linear programs that combines primal and
dual problems was proposed by K. V. Nguyen [1], where the pri
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 5, NO. 1, JANUARY 1994
96
Convergence Analysis of
Canonical Genetic Algorithms
Gunter Rudolph
Abstract- This paper analyzes the convergence properties of
the canonical genetic algorithm (CGA) with mutation, cross
ECE 580
Optimization Methods for
Systems and Control
Chapter 12
Recursive Least Squares
Stan Zak
March 01, 2013
ECE 580Optimization Methods for Systems and Control p. 1/16
Geometric Interpretation of the Least Squares
b
e=h-b
h
0
Range of A
h is closest t
ECE 580
Optimization Methods for
Systems and Control
Chapter 12
An Intro to the Least Squares
Stan Zak
February 27, 2012
ECE 580Optimization Methods for Systems and Control p. 1/1
The Least Squares MethodBackground
In January 1801, G. Piazzi briey observe
ECE 580
Optimization Methods for
Systems and Control
Chapter 11
An Introduction to Quasi-Newton Methods
Stan Zak
February 15, 2013
ECE 580Optimization Methods for Systems and Control p. 1/21
Idea Behind Quasi-Newton Methods
These methods lie somewhere int
ECE 580
Spring 2013
FunWork #2
Due on February 13
1. Exercise 7.12, page 129 from Textbook.
2. For the function
f (x1 , x2 ) = (x2 x1 )4 + 12x1 x2 x1 + x2 3,
(a) use MATLABs commands meshgrid and mesh to generate its 3D plot. The range of
x1 and x2 is the
c 2011 Society for Industrial and Applied Mathematics
SIAM J. OPTIM.
Vol. 21, No. 1, pp. 212230
A THREE-TERM CONJUGATE GRADIENT METHOD WITH
SUFFICIENT DESCENT PROPERTY FOR UNCONSTRAINED
OPTIMIZATION
YASUSHI NARUSHIMA , HIROSHI YABE , AND JOHN A. FORD
Abst
An Introduction to
the Conjugate Gradient Method
Without the Agonizing Pain
Edition 1 1
4
Jonathan Richard Shewchuk
August 4, 1994
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
Abstract
The Conjugate Gradient Method is the mos
ECE 580
Optimization Methods for
Systems and Control
Chapter 10
Conjugate Direction Methods
Stan Zak
February 13, 2013
ECE 580Optimization Methods for Systems and Control p. 1/10
Basic Conjugate Direction Algorithm
Let d(0) , d(1) , . . . , d(n1) be n Q-c
ECE 580
Optimization Methods for
Systems and Control
Chapter 9
Newtons Method and Conjugate Directions
Stan Zak
February 08, 2013
ECE 580Optimization Methods for Systems and Control p. 1/12
Newtons Method for a Function of n Variables
Assumption f C 2 imp
ECE 580
Optimization Methods for
Systems and Control
Chapter 8
Gradient MethodsAn Analysis
Stan Zak
February 06, 2013
ECE 580Optimization Methods for Systems and Control p. 1/10
The Method of Steepest Descent (SD) Applied to the Quadratic
The Method of St
ECE 580
Optimization Methods for
Systems and Control
Chapter 8
Gradient MethodsAn Introduction
Stan Zak
February 04, 2013
ECE 580Optimization Methods for Systems and Control p. 1/10
Review of the Gradient Properties
The direction of maximum increase of a
ECE 580
Optimization Methods for
Systems and Control
Chapter 7
Line (1D) Search Methodscontd.
Stan Zak
February 04, 2013
ECE 580Optimization Methods for Systems and Control p. 1/14
The Fibonacci Sequence Based Line SearchIntroduction
In our range reductio
ECE 580
Optimization Methods for
Systems and Control
Chapter 7
Line (1D) Search Methods
Stan Zak
January 28, 2013
ECE 580Optimization Methods for Systems and Control p. 1/18
The Objective Function
f :RR
ECE 580Optimization Methods for Systems and Control
ECE 580
Spring 2013
FunWork #1
Due on January 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.