代写MATH3161、代写Python，c++程序

日期：2022-03-11 11:29

THE UNIVERSITY OF NEW SOUTH WALES

SCHOOL OF MATHEMATICS AND STATISTICS

MATH3161/MATH5165–OPTIMIZATION

CLASS TEST 1

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

1 = {x ∈ R2 : x21 + 4x22 = 4, x1 ≥ 2x2 + 2};

2 = {x ∈ R2 : x21 + 4x22 ≤ 4, x1 ≥ 2x2 + 2}.

i) Sketch the feasible regions ?1 and ?2.

ii) Show that ?1 is not a convex set.

iii) Write ?2 in standard form.

iv) Show that ?2 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 4x31 + 3x21x22 ? 6x21

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.

Determine whether the function f on the convex set ? 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.