Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

SCHOOL OF MATHEMATICS AND STATISTICS
MATH3161/MATH5165–OPTIMIZATION
ONLINE CLASS TEST 2
Term 1, 2020
(1) TIME ALLOWED – 50 Minutes
(2) TOTAL NUMBER OF QUESTIONS – 3
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE NOT OF EQUAL VALUE
(5) ALL STUDENTS MAY ATTEMPT ALL QUESTIONS. MARKS GAINED ON ANY
QUESTION WILL BE COUNTED.
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
(7) ALL STUDENTS WILL NEED TO SCAN (TAKE PHOTO OF) EVERY
PAGE OF THEIR WORKINGS, CONVERT IT TO PDF AND UPLOAD
ONE PDF FILE TO MOODLE WITHIN 90 MINUTES
All answers must be written in ink. Except where they are expressly required pencils may only
be used for drawing, sketching or graphical work.
1. [12 marks] Consider the following constrained optimization problem
(P1) Minimize
x∈R2
x1x2
subject to
√
5 x1 +
√
5 x2 − 1 ≤ 0, 10− x21 − x22 = 0.
Let x∗ = [−√5,√5]T .
i) Show that x∗ is a regular feasible point for the problem (P1).
ii) Show that x∗ is a constrained stationary point for the problem (P1).
iii) Using the second-order sufficient optimality conditions, show that x∗ is a strict local
minimizer for the problem (P1).
2. [16 marks] Consider the optimization problem
(EP) Minimize
x∈Rn
n∑
i=1
x2i
subject to
n∑
i=1
xi = c,
where c ∈ R, x = [x1, x2, . . . , xn]T ∈ Rn and n > 1.
i) Show that the problem (EP) is a convex optimization problem.
ii) Find a constrained stationary point x∗ of the problem (EP).
iii) Show that the point x∗ in part ii) is a global minimizer of (EP).
iv) Hence or otherwise, show that
n
n∑
i=1
x2i ≥
(
n∑
i=1
xi
)2
,
for any given real numbers x1, x2, . . . , xn.
3. [12 marks] Consider applying the method of steepest descent with exact line searches
to the strictly convex quadratic function
f(x) =
1
2
xTAx + bTx + d,
where A is a positive definite n× n symmetric constant matrix, b is a constant n× 1
vector and d is a scalar. Suppose that the starting point x(1) can be expressed as x(1) =
x∗ + v, where v is an eigenvector of A with eigenvalue λ and x∗ is the minimizer of f .
i) Show that the initial steepest descent direction is s(1) = −λv.
ii) Using the line search condition, find the exact minimizer α(1) of f(x(1) + αs(1)).
iii) Determine whether or not the steepest descent method terminates in one iteration.