UNIVERSITY OF MELBOURNE
SCHOOL OF MATHEMATICS AND STATISTICS
MAST30013 Techniques in Operations Research Semester 1, 2023
Assignment 2 Due: 5 pm, Mon, 17 April
- Solution must be typeset in LaTex.
- Please submit your solution in Canvas by the due date.
- Show all necessary working.
Consider the function f : R4 → R:
f(x) = 100(x21−x2)2+(x1−1)2+(x3−1)2+90(x23−x4)2+10.1((x2−1)2+(x4−1)2)+19.8(x2−1)(x4−1)
You are required to do a computational study comparing the below three methods for finding
stationary points and global minima of f .
i Steepest descent method;
ii Newton’s method;
iii BFGS Quasi-Newton method.
1. Create a set of instances which consists of 1000 randomly generated initial points for the
algorithms. Test the algorithms on the instance set and compare their average performance
in terms of solutions found and computational time. Use the following parameters:
• tolerance 1 = 10−2 for the three methods,
• tolerance 2 = 10−5 for the Golden section search,
• step size T = 10,
• initial points with coordinate values range xi ∈ (−10, 10), i = 1, 2, 3, 4.
(a) You should report the average performance of your algorithms in tables. For example:
f value Stationary point No. of times found Ave iterations
per search
Ave time per
search (sec)
−2 (x1, x2, x3, x4) 6 7.8 0.2157
...
(b) Classify any stationary points found as local/global optima or saddle points.
(c) Discuss, in words, the conclusion you have arrived at from your computational study.
2. Discuss the behaviour of Newton’s method for optimising f with starting point (−3,−1,−3,−1).
You need to modify the given code (f.m, GRADF.m, HESS.m and script.m) from Canvas in
order to take function f as input. Only include a screenshot of the Matlab script.m in your LaTex
submission. Do NOT include the other Matlab code.