1. (25 marks) True of False (No justication necessary)
(a) The set f(x; y) : y = 2x + 1 for (x; y) 2 R2g is a subspace of R2.
(b) The set f(x; y) : y = x2 for (x; y) 2 R2g is a subspace of R2.
(c) Recall the function kxk0 is dened as the number of nonzero entries in x 2 Rn. The
function kxk0 a valid norm.
(d) For unconstrained optimization problem, a stationary point with a PSD hessian ma-
trix must be a local minimizer.
(e) For unconstrained optimization problem, a stationary point without a PSD hessian
matrix can not be a local minimizer.
2. (25 marks) Let A = uuT 2 R1010 be a symmetric matrix, where u =
2
6664
1
2
…
10
3
7775
2 R10:
(a) Compute rank(A).
(b) Compute kAk1.
(c) Compute kAkF .
(d) Compute tr(A).
(e) Compute det(A).
3. (25 marks) Given A 2 Rnn; b 2 Rn and > 0, let f(x) = 1
2kAx bk22
+
2 kxk22
.
(a) Compute rf(x).
(b) Compute r2f(x).
(c) Prove r2f(x) is PD for any x 2 Rn.
(d) Find the stationary point of f(x).
(e) Using the sucient conditions, show this stationary point is a local minimizer.
4. (25 marks) Given two vectors a; b 2 Rn, let f(x) = 1
2kx ak22
+ 1
2kx bk22
.
(a) Compute rf(x).
(b) Compute r2f(x).
(c) Find the stationary point of f(x).
(d) Using the sucient conditions, show this stationary point is a local minimizer.
(e) Using the denition of global minimizer, show this stationary point is also a global
minimizer. (hint: completing squares)
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