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
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
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
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