620-361 Operations Research Techniques and
Algorithms
Practice Class 1
1 (a) The Fibonacci numbers Fn satisfy the recursion
Fn = Fn1 + Fn2
(1)
with F0 = F1 = 1.
Show that1
1 1+ 5
Fn =
2
5
n+1
1 5
2
n+1
.
(2)
(b) Now let n = Fn1 /Fn . Use (1) to derive a

620-361 Operations Research Techniques and
Algorithms
Practice Class 1 Solutions
1. (a) This is given in the course notes but we elaborate more here. Because
of the form of the Fibonacci recurrence, it makes sense to try a series
of the form Fn = n (compa

620-361 Operations Research Techniques and
Algorithms
Practice Class 3
1. Let f : n , f C 2 . Consider a stationary point x n and let
1 , 2 , . . . , n be the eigenvalues of 2 f (x ). Suppose that 1 < 0 and
2 > 0. Show that x is neither a local minimum no

620-361 Operations Research Techniques and
Algorithms
Practice Class 5
1. How can you tell if a matrix is positive denite, positive semi-denite,
negative denite or negative semi-denite?
2. (a) Name two ways to calculate, from a given matrix, whether it is

620-361 Operations Research Techniques and
Algorithms
Practice Class 4
You are playing cards with your friends when you notice that the oor of
the room is slanted. The slant is sucient to provide additional viewing of
your friends cards according to the f

620-361 Operations Research Techniques and
Algorithms
Practice Class 8
1. Write down the Wolfe dual of:
x1 x2 + x2
2
min
n
x
s.t.
x2
1
4
x0
+ x2 1
2
Reduce the number of variables of the dual problem. [Hint: Eliminate the
vector of multipliers correspondi

620-361 Operations Research Techniques and
Algorithms
Practice Class 6
Consider the constrained nonlinear program
1 2 1 2
x + x x1 + x2
2 1 2 2
x1 , x2 0.
min f (x) =
x2
subject to
1. You can see that this is a 2-D nonlinear optimisation problem with ineq

620-361 Operations Research Techniques and
Algorithms
Practice Class 2
1 Consider the unconstrained nonlinear program:
min f (x) =
2
x
4 3
x x1 x2 8x2 + 3x2 .
2
2
3 1
5000
0
5000
10
5
10
5
0
0
5
5
10
10
(a) Show that x = (1, 2)T is a stationary point of f

620-361 Operations Research Techniques and
Algorithms
Practice Class 7
1. Consider the nonlinear program (NLP):
min
n
f (x)
s.t.
gi (x) 0, i = 1, . . . , m.
x
Prove that if (NLP) is a convex program with KKT point (x , ), then
x minimizes the Lagrangian f

620-361 Operations Research Techniques and
Algorithms
Practice Class 5
1. How can you tell if a matrix is positive denite, positive semi-denite,
negative denite or negative semi-denite?
Solution:
(a) Positive denite - we say a matrix, M , is positive deni