MA208 Optimisation Theory
Solutions to Exercises 1 (Real numbers)
Solution to Exercise 1.1.
Let A have n elements, A = cfw_ a1 , a2 , . . . , an . We prove the claim by induction on n. If
n = 1, then a = a1 because for a A we have a = a1 and therefore a a
MA208 Optimisation Theory
Solutions to Exercises 3 (Algorithms, BellmanFord)
Solution to Exercise 3.1.
When the array elements are S[1], S[2], S[3], S[4], S[5], S[6] = 5, 3, 3, 4, 3, 8, then their
minimum is clearly m = 3. The algorithm also returns i = 2
MA208 Optimisation Theory
Solutions to Exercises 2 (Digraphs)
Solution to Exercise 2.1.
(a) For a directed graph D = (V, A), the arcs are a set of ordered pairs (u, v) with
u, v V, u = v. Because A is a set, no pair appears more than once. So the
maximum
MA208 Optimisation Theory
Exercises 6 (Functions on R2)
Exercise 5.1.
Consider the function f : R2 R, mentioned in the lecture, defined by
0
if ( x, y) = (0, 0)
xy
f ( x, y) =
otherwise.
2
x + y2
(a) Show that f is not continuous at (0, 0).
(b) Show that
2014 examination Solutions
Answers MA208, Optimisation Theory
Each question gives 25 marks.
1
(a) ( 8 marks )
The Weierstrass Theorem cannot be applied directly because the domain [0, ) of the function f is not compact. If f (x) 0 for all x 0, then f triv
2015 examination Solutions
Answers MA208, Optimisation Theory
Each question gives 25 marks.
1
Skills tested: Understanding of basic concepts about closed and open sets and applying them in
new contexts. Understanding and applying the Bellman-Ford algorith
Summer 2013 examination
MA208
Optimisation Theory
Half Unit
1 Suitable for all candidates
Instructions to candidates
Time allowed: 2 hours
This paper contains 5 questions. You may attempt as many questions as you wish, but only your BEST 4
answers will co
Summer 2014 examination
MA208
Optimisation Theory
(Half Unit)
Suitable for all candidates
Instructions to candidates
Time allowed: 2 hours
This examination paper contains 5 questions. You may attempt as many questions as you wish,
but only your best 4 ans
Solutions for the MA208 Exam (Summer 2013)
1
(a) (i) (Essentially bookwork; ) The rst version of the Bellman-Ford algorithm computes exactly these d(v; i). The algorithm (without checking for a negative cycle)
is as follows:
1.
set d(s; 0) = 0;
2.
v V , v
Summer 2015 examination
MA208
Optimisation Theory
(Half Unit)
2014/2015 syllabus only not for resit candidates
Instructions to candidates
Time allowed: 2 hours
This examination paper contains 5 questions. You may attempt as many questions as you wish,
but