MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #3
Last modied: October 9, 2007
Due Thursday, Oct. 11 at class
This problem set has been shortened to allow more time for watching baseball
playos.
1. Showing that a norm t
Math 116 - Problem Set 4
(Thanks to Nick Wage for his write-up. Also, the solution to problem 4c have not been
typed up yet. A new version will be posted in the future).
1.
(a) Let x0 (0, 1] and > 0. Then take 0 < < mincfw_
x2 x0 1x0
0
, 2 , 2 .
2
Then if
Math 116 Problem Set 2
October 10, 2007
(Thanks to Ameya Velingker for his write-up.)
1. (a) This modication does not make our system a vector space. Notice that Observe that if the
distributive law were to hold, we would have = 0S = (1 + (1)S = 1 S + (1)
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Proofs to learn for Quiz 1 on October 11.
Two of these, chosen at random, will appear on the quiz.
Last modied: October 9, 2007
1. Prove that the interior of a convex set C is convex.
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Proof List for Quiz 2 on November 8
Last modied: November 4, 2007
1. Prove that the Banach space lp for 1 p < is separable.
2. Suppose that X is a Hilbert space and that M is a closed
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Improved statements of proofs for Quiz 2
Last modied: November 7, 2007
Notes in parentheses were not added to the text used for the quiz. They
address issues that were raised in email
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Proof List for Quiz 3 on December 6
Last modied: December 3, 2007
As usual, the quiz will have one proof chosen at random from the rst four
and another chosen at random from the last
MATHEMATICS 116, FALL 2007-8
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Proof List for the Final Examination
Last modied: January 13, 2008
The exam will contain one proof chosen at random from each set of three:
1-3, 4-6, .
1. Prove that the interior of
Mathematics 116. Problem Set 1 Solutions
Friday, 28 September 2007
1. We require x, y 0. The ingredient restrictions are
3x + y 15 (chocolate bars)
x + y 7 (pounds sugar)
x + 2y 12 (pounds our)
These restriction dene a convex set with vertices at A = (0,
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Outline #1 (Generalizing from two dimensions)
Last modied: September 10, 2007
Reading. Luenberger, Chapter 1
Lecture topics.
1. Why such a long introduction?
The general approach in L
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #1
Last modied: September 26, 2007
Due Thursday, Sept. 27 at class
1. The Math 116 bakery has started producing chocolate goods.
A batch of fudge squares uses 3 bars of cho
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #2
Last modied: October 3, 2007
Due Thursday, Oct. 4 at class
1. (a) Suppose that you modify the NSA budget system from the lecture
outline by dening 0S = so that the axiom
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Outline #2 (Vector Spaces)
Last modied: October 4, 2007
Report errors by email to bamberg@tiac.net
Reading. Luenberger, Section 2.1 - 2.9
Lecture topics.
1
1. Axioms for a Vector Spac
MATHEMATICS 116, FALL 2007-2008
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #12
Last modied: January 4, 2008
Due Monday, January 14 before 5PM, in Charles Chens mailbox on the 3rd oor.
This deadline is rm, since solutions will be posted then.
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #11
Last modied: December 12, 2007
Due Tuesday, December 18 at class. If you are leaving town early, deliver it
early to Nike. Work the last two problems before Thursday, s
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Outline #4 (Hilbert Space)
Last modied: October 22, 2007
Reading. Luenberger, Chapter 3, omitting any consideration of complex vector
spaces.
Lecture topics.
1. Axioms for an inner pr
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #4
Last modied: October 12, 2007
Due Thursday, Oct. 18 at class
This is short enough that you might have time left over to watch the baseball
playos.
1. (a) Show that the f
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Outline #5 (Dual Spaces and the Hahn-Banach Theorem)
Last modied: November 1, 2007
Reading. Luenberger, Chapter 5, sections 5.1 - 5.4 and section 5.12. These
sections of Chapter 5 are
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #5
Last modied: October 20, 2007
Due Thursday, Oct. 25 at class.
1. (a) Carry out one more step of the example of the Gram-Schmidt process
for the space L2 [1, 1]that was b
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Outline #6 (Applications of the Hahn-Banach Theorem)
Last modied: November 20, 2007
Reading. Luenberger, Chapter 5, sections 5.5 - 5.9 and 5.11-5.13. None of this
is relevant to the q
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #6
Last modied: October 26, 2007
Due Thursday, Nov. 1 at class if the World Series ends in 4 or 5 games.
Due Saturday, Nov. 3 at 5PM if the Series continues to Wednesday ni
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Outline #7 (Calculus of Variations)
Last modied: December 6, 2007
Reading. Luenberger, Chapter 7, sections 7.1-7.5 and 7.7.
1
Lecture topics.
1. Review of the two-variable case
Let f
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #7 (revised)
Last modied: November 7, 2007
Due Thursday, Nov. 8 at class.
1. A convex linear functional on X is a sublinear functional that meets the
one additional require
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Outline #8 (Convex Functionals)
Last modied: December 2, 2008
Reading. Luenberger, Chapter 7, sections 7.8 and 7.10-7.12. We are skipping
the starred sections 7.9 and 7.13.
Lecture to
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #8
Last modied: November 14, 2007
Due Tuesday, Nov. 20 at class.
1. Luenberger, Problem 5.14.1.
2. Luenberger, Problem 5.14.3. Explain why this result shows that the dual
o
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #9
Last modied: November 25, 2007
Due Thursday, Nov. 29 at class.
1. Luenberger, Problem 5.14.6. This is like example 2 of section 5.9.
Hint (based on discussions with Dara
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Outline #9 (Inequality Constraints)
Last modied: December 18, 2007
Reading. Luenberger, Chapter 8. If you want to understand the term projects,
you should work through these simple ex
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #10
Last modied: December 3, 2007
Due Thursday, December 6 at class. All these problems are relevant to the quiz
on that date.
1. Luenberger, Problem 7.14.1. This is an int
MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Outline #3 (Banach Spaces)
Last modied: October 4, 2007
Reading. Luenberger, Sections 2.10 - 2.15
Lecture topics.
1. Another Cauchy-Schwarz proof
The so-called Cauchy-Schwarz inequali