TERM 1, 2020
(1) TIME ALLOWED – 50 Minutes
(2) TOTAL NUMBER OF QUESTIONS – 4
(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. GRADES OF DISTINCTION AND HIGH DIS-
TINCTION WILL REQUIRE SATISFACTORY PERFORMANCE ON ALL QUES-
TIONS, INCLUDING STARRED QUESTIONS
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
All answers must be written in ink. Except where they are expressly required pencils may
only be used for drawing, sketching or graphical work.
MATH3161/MATH5165–OPTIMIZATION CLASS TEST 1 Page 2
1. [8 marks] Consider the feasible regions of two optimization problems
Ω1 = {x ∈ R2 : x21 + 4×22 = 4, x1 ≥ 2×2 + 2};
Ω2 = {x ∈ R2 : x21 + 4×22 ≤ 4, x1 ≥ 2×2 + 2}.
i) Sketch the feasible regions Ω1 and Ω2.
ii) Show that Ω1 is not a convex set.
iii) Write Ω2 in standard form.
iv) Show that Ω2 is a convex set. State any results that you use.
2. [14 marks] Consider the problem of minimizing the function f
f(x) = 2x31x
2
2 − 4×31 + 3x21x22 − 6×21
on R2.
i) Calculate the gradient ∇f(x) and the Hessian ∇2f(x) of f .
ii) Show that x∗α =
[
0
α
]
is a stationary point of f on R2 for each α ∈ R .
iii) Find the other three stationary points of f on R2.
iv) Identify, as far as possible using Hessian information, the three stationary points
of f of part iii) as local minimizers, local maximizers or saddle points, etc.
v) Determine whether or not the stationary point x∗α of part ii) is a local minimizer
of f for |α| < √2.
3. [4 marks] Consider the function f(x1, x2) = e
− 1
2
(x21+x
2
2) on the convex set Ω where
Ω = {(x1, x2) ∈ R2 | x21 + x22 < 1}. The Hessian ∇2f(x1, x2) of f is given by
∇2f(x1, x2) = e− 12 (x21+x22)
[
x21 − 1 x1x2
x1x2 x
2
2 − 1
]
.
Determine whether the function f on the convex set Ω is convex, strictly convex,
concave, strictly concave or neither.
4. [4 marks] Let G be an n × n symmetric matrix with the spectral decomposition
G = QTDQ, where Q is an n × n orthogonal matrix (i.e. QTQ = QQT = I),
D = diag(λ1, . . . , λn) is an n×n diagonal matrix whose diagonal elements, λ1, . . . , λn,
are the eigenvalues of the matrix G, and I is an n× n identity matrix.
i) Show that if G is positive semi-definite then λi ≥ 0, for i = 1, 2, . . . , n.
ii) Show that if λi > 0, for i = 1, 2, . . . , n, then G is positive definite.
NOTE: For all questions you must show all your working and give reasons for
all statements that you make.
END OF EXAMINATION
