Solutions for Homework 4 Optimization
Spring 2012
Problem 4.1
Problem 26 on page 260 of the textbook.
Solution:
For Steepest Descent we have the result
e(k+1)
A
1 (k)
e
+1
A
1
+1
k+1
e(0)
A.
6 places accuracy implies relative error of 106 if the size of
Solutions for Homework 3 Optimization
Spring 2012
Problem 3.1
Problem 15 on page 259 of the textbook.
Solution: We have for > 0
xk+1 = xk + k rk , (1 )k k (1 + )k
k =
T
rk rk
T
rk Ark
First note that if = 1 then = 0 and there is no progress toward x = A1
Solutions for Homework 2 Optimization
Spring 2012
Problem 2.1
Let A Rnn be a symmetric positive denite matrix, C Rnn be a symmetric nonsingular
matrix, and b Rn be a vector. The matrix M = C 2 is therefore symmetric positive denite.
Also, let A = C 1 AC 1
Homework 5 Optimization
Spring 2012
The solutions will be posted on Monday, 3/26/12
Problem 5.1
Problem 1 on page 28 of the textbook.
Solution: Using slack and surplus variables yields the conversion
min x + 2y + 3z
subject to
x+y 2
x+y 3
x+z 4
x+z 5
x0
y
Homework 1 Optimization
Spring 2012
Problem 1.1
Let f (x) = x3 3x + 1. This polynomial has three distinct roots.
(1.1.a) Consider using the iteration function
1
1 (x) = (x3 + 1)
3
Which, if any, of the three roots can you compute with 1 (x) and how would
Solutions for Homework 4 Optimization
Spring 2012
Problem 4.1
Problem 26 on page 260 of the textbook.
Solution:
For Steepest Descent we have the result
e(k+1)
A
1 (k)
e
+1
A
1
+1
k +1
e(0)
A.
6 places accuracy implies relative error of 106 if the size of
Solutions for Homework 3 Optimization
Spring 2012
Problem 3.1
Problem 15 on page 259 of the textbook.
Solution: We have for > 0
xk+1 = xk + k rk , (1 )k k (1 + )k
k =
T
rk rk
T
rk Ark
First note that if = 1 then = 0 and there is no progress toward x = A1
Solutions for Homework 2 Optimization
Spring 2012
Problem 2.1
Let A Rnn be a symmetric positive denite matrix, C Rnn be a symmetric nonsingular
matrix, and b Rn be a vector. The matrix M = C 2 is therefore symmetric positive denite.
Also, let A = C 1 AC 1
Homework 1 Optimization
Spring 2012
Problem 1.1
Let f (x) = x3 3x + 1. This polynomial has three distinct roots.
(1.1.a) Consider using the iteration function
1
1 (x) = (x3 + 1)
3
Which, if any, of the three roots can you compute with 1 (x) and how would
Optimization Final Exam
In-class Exam
Open Notes, Textbook, Homework Solutions Only
Calculators Allowed
Due beginning of Class Monday, April 16, 2012
Question
Points
Points
Possible Awarded
1. Basics
20
2. Projected Gradient
20
3. Quadratic Program
30
4.
Optimization Exam 2
Take-home Exam
Open Notes, Textbook, Homework Solutions Only
Calculators Allowed
Due beginning of Class Monday, April 16, 2012
Question
Points
Points
Possible Awarded
25
25
1. Geometry
2. Geometry and
Optimization
3. Simplex Method
4.
'
$
Set 20: Constrained Optimization on Rn
Part 8
Kyle A. Gallivan
Department of Mathematics
Florida State University
Optimization
Spring 2012
&
1
%
'
$
Nonlinear Program
min f (x)
subject to:
hi (x) = 0, i E
gi (x) 0, i G
L(x, ) = f (x) +
hi (x)i +
iE
iG
Optimization Exam 2
Take-home Exam
Open Notes, Textbook, Homework Solutions Only
Calculators Allowed
Due beginning of Class Monday, April 16, 2012
Question
Points
Points
Possible Awarded
25
25
1. Geometry
2. Geometry and
Optimization
3. Simplex Method
4.
Homework 5 Optimization
Spring 2012
The solutions will be posted on Monday, 3/26/12
Problem 5.1
Problem 1 on page 28 of the textbook.
Solution: Using slack and surplus variables yields the conversion
min x + 2y + 3z
subject to
x+y 2
x+y 3
x+z 4
x+z 5
x0
y