Math 164, Lecture 2, Vese
Homework #9, due on Friday, March 17,, OR on Monday, March 20
(no late homework accepted after March 20)
Notes:
REMINDER: Final exam on Monday, March 20, 2005, time 3:00pm-6:00pm.
- Sample final practice problems are posted on the class web page.
- All sections and topics are covered for the final. However, more questions will be from the second part
(already covered after the midterm).
Sections covered for the final exam:
- 2.2, 2.3, 3.1, 4.1-4.4, 5.2 (except 5.2.1), 5.2.2 (already covered for the midterm)
- 6.1, 6.2 (proof of Thm. 6.2 not included), 6.2.1,
- Appendices A6, B4, B5, B6, B7.
- 2.3.1, 2.6, 2.7 (except Thm. 2.2), 2.7.1, 3.2
- 10.2, 10.3 (except Thm. 10.1)
- 14.2, 14.3 (only what is presented on page 437, not the discussion on the perturbed problem)
- 14.4 (read also Lemma 14.5, but this Lemma is not included for the final)
- 14.5.1 (just to know the sufficient conditions, and apply them to a specific example)
Problems:
[1] Consider the problem
minimize f(x) = 1
2xTQx −cTx
where Qis a positive definite matrix. Prove that Newton’s method will determine the minimizer of fin one
iteration, regardless of the starting point (use Appendix B5 and Thm. 2.1, page 22)
[2] Consider the problem
minimize f(x) = x2
1+x2
1x2
3+ 2x1x2+x4
2+ 8x2
subject to 2x1+ 5x2+x3= 3.
(a) Determine which of the following points are stationary points:
(i) (0,0,2)T; (ii) (0,0,3)T; (iii) (1,0,1)T
(b) Determine whether each stationary point is a local minimizer, a local maximizer or a saddle point.
[3] Solve the problem
maximize f(x) = x1x2x3
subject to x1
a1
+x2
a2
+x3
a3
= 1 (a1, a2, a3>0)
[4] Solve the problem: Minimize f(x1, x2) = 1
2x2
1+x2
2subject to
2x1+x2≥2
x1−x2≤1
x1≥0.
1