MA 321 : Nonlinear Recitation 2
Optimization
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Signature:
Your solutions should be fully justified. Solutions will be graded for correctness and quality of explanations.
1. Show that if A, B are n → n positive semidefinite matrices, then their sum A + B is also positive semidefinite.
2. Find the global minimum and maximum points of the function f (x, y) = x2 + y 2 + 2x ↑ 3y over the closed unit ball
B[0, 1].
3. Find the global minimum and maximum points of the function f (x, y) = 2x ↑ 3y over the set
S = {(x, y) : 2x2 + 5y 2 ↓ 1}.
4. • Let A be n → n symmetric matrix. Show that A is positive semidefinite if and only if there exists a matrix
B ↔ Rn→n such that A = BB T
• Let x ↔ Rn and A is defined as
Aij = xi xj
i, j = 1, 2, 3...n. Show that A is positive semidefinite and that it is not a positive definite matrix when n > 1.
5. For the following matrices determine whether they are positive/negative semidefinite/definite or indefinite:
2 2 2
(a) 2 3 3
2 3 3
↑5 1 1
(b) 1 ↑7 1
1 1 ↑5
2 2 0 0
2 2 0 0
(c)
0
0 3 3
0 0 3 3
6. For each of the following functions, find all the stationary points and classify them according to whether they are
saddle points, strict/nonstrict local/global minimum/maximum points:
(a) f (x1 , x2 ) = (4x21 ↑ x2 )2
(b) f (x1 , x2 ) = x41 + 2x21 x2 + x22 ↑ 4x21 ↑ 8x1 ↑ 8x2
,
Optimization
Honor Pledge: I have abided by the Honor Code for this assignment.
Signature:
Your solutions should be fully justified. Solutions will be graded for correctness and quality of explanations.
1. Show that if A, B are n → n positive semidefinite matrices, then their sum A + B is also positive semidefinite.
2. Find the global minimum and maximum points of the function f (x, y) = x2 + y 2 + 2x ↑ 3y over the closed unit ball
B[0, 1].
3. Find the global minimum and maximum points of the function f (x, y) = 2x ↑ 3y over the set
S = {(x, y) : 2x2 + 5y 2 ↓ 1}.
4. • Let A be n → n symmetric matrix. Show that A is positive semidefinite if and only if there exists a matrix
B ↔ Rn→n such that A = BB T
• Let x ↔ Rn and A is defined as
Aij = xi xj
i, j = 1, 2, 3...n. Show that A is positive semidefinite and that it is not a positive definite matrix when n > 1.
5. For the following matrices determine whether they are positive/negative semidefinite/definite or indefinite:
2 2 2
(a) 2 3 3
2 3 3
↑5 1 1
(b) 1 ↑7 1
1 1 ↑5
2 2 0 0
2 2 0 0
(c)
0
0 3 3
0 0 3 3
6. For each of the following functions, find all the stationary points and classify them according to whether they are
saddle points, strict/nonstrict local/global minimum/maximum points:
(a) f (x1 , x2 ) = (4x21 ↑ x2 )2
(b) f (x1 , x2 ) = x41 + 2x21 x2 + x22 ↑ 4x21 ↑ 8x1 ↑ 8x2
,
