Analysis and Optimization. Practice nal.
Closed book, notes. No calculators or electronic devices.
Explain and justify wherever possible.
1. (a) Let f : Rn R be a C 2 function which attains its maximum at 0. What can you
say about the gradient and the Hes
Analysis and Optimization. Final exam. 100 points.
Closed book, notes. No calculators or electronic devices.
Explain and justify wherever possible.
1. (a) Let f : Rn R be a C 2 function which attains its maximum at 0. What can you
say about the gradient a
Analysis and Optimization. Practice midterm. (short) solutions
1. (a) Dene: f : Rn R a convex function. [4 points]
(b) Quote the second-order Taylors formula for f : Rn R a C 2 function. [4 pts]
2. True/False (no justication required). Here, f : Rn R is a
Analysis and Optimization. Midterm. 40 points.
Explain and justify wherever possible.
1. what is.
(a) the Jacobian of a C 1 function f : Rn Rm at x Rn . [2 points]
(b) a convex set in Rn [2 pts]
2. True/False (no justication required). Here, f : Rn R is a
Analysis and Optimization. Practice midterm.
1. (a) Dene: f : Rn R a convex function. [4 points]
(b) Quote the second-order Taylors formula for f : Rn R a C 2 function. [4 pts]
2. True/False (no justication required). Here, f : Rn R is a C 2 function. [10
Analysis and Optimization. HW set 7. Explain and justify wherever possible.
1. (not for credit). 3.1.1, 3.1.2 (see example 2), 3.1.5, 3.2.2, 3.2.3
2. Let f (x, y) = cos(x)3 sin(y). Find the critical points of f . Which of these are extreme
points ?
3. Let
Analysis and Optimization. HW set 2. Explain and justify wherever possible.
Not for credit. I.3.1,3; I.4.2,4; I.5.1,4.
1. Find the rank of
2. Find all solutions of
1 0 1 2
A = 2 1 1 1
3 2 1
0
1
0
0
1
1
1
0
0
0
w
x
0
1 y
1
z
0
1
1
0
b1
b2
=
b3
b4
Analysis and Optimization. HW set 4. Explain and justify wherever possible.
1. (not for credit). 2.1.1,2.1.2,2.1.7,2.2,2.3,2.5,2.6,3.1,3.4,3.6
2. Let F (x, y) = xy. Sketch the level sets of F for the levels = 1, 0, 2; and the gradient
of F at the points (
Analysis and Optimization. HW set 6. Explain and justify wherever possible.
1. (not for credit). 2.7.1,2.7.2,2.7.4,2.7.6
2. 2.7.9
3. Consider the subset S of R3 dened by:
x2 + 2y 2 + z 4 xy 2yz = 1
Show that there is a C 1 function g : R2 R dened near (1,
Analysis and Optimization. HW set 5. Explain and justify wherever possible.
1. (not for credit). II.4.5,4.6,6.1,6.2
2. Let x1 , . . . , xn be real numbers. Using Jensens inequality, show that
(x1 + + xn )4 n3 (x4 + + x4 )
1
n
3. Show that f (x, y) = exp(x
Analysis and Optimization. HW set 3. Explain and justify wherever possible.
1. (not for credit). 1.6.1, 1.6.3, 1.6.4,1.7.1,1.7.3,1.7.6
2. Diagonalize
A=
1 4
1 3
3. Find the eigenvectors of
0 1 0
A = 0 0 1
0 0 0
Is it diagonalizable ?
4. Consider the quadr
Analysis and Optimization. HW set 1. Explain and justify wherever possible.
Not for credit: I.2.1,2,5.
1. Let
1 2
1 3
A=
and
2 1
0 1
B=
Evaluate 2A B, AB, BA, A , A1 , det(A).
2. For which numbers x are the vectors
1
x
and
2x
1
linearly independent ?
3. L