SCHOOL OF MATHEMATICS AND STATISTICS
MAST30013 Techniques in Operations Research Semester 1, 2023
Assignment 3 Due: 5pm, Wed, 10 May
• All submissions must be through Canvas and typeset in LaTex.
• Show all necessary working.
1. Consider the program
min −(x1 − 1)2 − (x2 − 2)2
s.t. x1 ≥ 0
x2 ≥ x1
(a) Using the graphical method, find all local minimums and maxi-
mums of the program. Also, discuss the global min.
(b) State and solve the KKT conditions of the program, and compare
to your results from the previous question.
(c) Confirm the local minimality or maximality of the KKT points
using the second order condition.
2. Consider the program
min 1
2
‖t− x‖2
s.t. Ax = 0
where t, x ∈ Rn and A is an m× n matrix of rank m, where m < n.
(a) Interpret the problem as a geometric problem.
(b) State and solve the KKT conditions of the program, and hence
find the global minimum.